# Methods for proving an equivalence relation

I'll be taking introductory abstract algebra in the fall, and so to prepare, I'm working through Pinter's text. Chapter 12 includes a number of exercises asking the student to prove that something is an equivalence relation and to describe the associated partition. For example: In $\mathbb{Q}$, $r \sim s$ iff $r - s \in \mathbb{Z}$.

I think I'm okay with most (?) of this. I'd show this is an equivalence relation like so: If $x, y, z \in \mathbb{Q}$, then $x - x = 0 \in \mathbb{Z}$, and so $x \sim x$. Second, if $x \sim y$, then $y - x = -(x - y) \in \mathbb{Z}$, and so $x \sim y \implies y \sim x$. Finally, if $x \sim y$ and $y \sim z$, then $x - z = (x - y) - (z - y) \in \mathbb{Z}$, and so $x \sim z$.

The two parts I'm not sure about: First, that last f on the iff. Essentially, this means I have to prove that if $r \sim s$, then their difference is an integer, right? But I thought this is merely how this particular equivalence relation is defined. How do I know that $r \sim s$ until I look at their difference?

Second, I have a basic idea of what the partition is, but I'm not sure how to form the statement. The equivalence class $[q] = \{k + q : k \in \mathbb{Z}, q \in \mathbb{Q} \}$. But if anything, that seems as though it would be the definition for a single equivalence class, not the description of the partition. (Now that I look at it again, it also leaves out the fact that $k$ is arbitrary but $q$ is fixed.)

If anyone can offer any hints, I'd very much appreciate it. (Though I'm guessing the second of my questions might be more amenable to a No-this-is-how-you-do-it than to a hint, per se.)

-
The text defines $r$ to be equivalent to $s$ iff (if and only if) the difference is an integer. There is nothing for you to prove, the difference being an integer is the defining condition for the equivalence relation. – André Nicolas Aug 7 '13 at 4:32
@AndréNicolas So are you saying the question is poorly written? I guess I was originally reading it like "r R s <--> r - s in Z," where R is not necessarily an equivalence relation. But ~ usually does mean equivalence relation, right? – dmk Aug 9 '13 at 14:53
The question is not poorly written, it is quite clear. But to show this is an equivalence relation, all you need to do is to verify that the conditions for an equivalence relation are satisfied. You don't have to prove that if $r\equiv s$ then $r-s$ is an integer, you have been told that is what $\equiv$ means for this exercise, – André Nicolas Aug 9 '13 at 14:59
@AndréNicolas So what you were saying then is that there's nothing for me to prove beyond what I'd already proved? – dmk Aug 9 '13 at 15:40
Yes, you had completely finished proving that the relation is an equivalence realtion. And it is a clean well-organized proof. – André Nicolas Aug 9 '13 at 16:40

Hint

Prove that the associated partition is $$\{[q]\, |\, q\in[0,1)\}$$

-
My first thought was, OK, that's true, but is it necessary to be so restrictive? For example, in Z_4, [1] = [5] = [61] = ... Well, sure, but we only need [1]. Two days later, though, it clicked: "Succinct" is a better adjective than "restrictive." (And to complete the analogy for myself, the partition of Z_4 would be {[k] : k = 0, 1, 2, 3}.) (Not sure how to do the @ thing here...) – dmk Aug 9 '13 at 15:16
If we want to give a set usually we write its elements without repetition. – user63181 Aug 9 '13 at 15:28

For your first part: Definitions are iff, or if and only if statements, as they are essentially stating that two things are equivalent -- the new term, and its definition. It shouldn't be affecting what you need to prove, except that you can use both that $r \sim s \implies r - s \in \mathbb{Z}$ and that $r - s \in \mathbb{Z} \implies r \sim s$.

For the second part, a better way of putting it would be as follows:

For each $q \in \mathbb{Q}$, the equivalence class of $q$ under $\sim$ is as follows: $[q] = \{q + k : k \in \mathbb{Z}\}$.

By saying this, you describe all of the equivalence classes making up the partition in one statement, and also take care of having $q$ fixed and $k$ varying.

-

This post is to answer the question above:

How do I know that r∼s until I look at their difference?

Answer: Use an indexing number m, with m = r - g(r), with g(r) being the next lower integer from r. Then each class can be represented as Am (I don't know how to subscript). m will be some rational number between zero (inclusive) and 1 (non-inclusive).

-


# 5 Vertices Denoted by K5 part B

#### Joystar1977

##### Active member
Consider the complete graph with 5 vertices, denoted by K5.

B. How many edges are in K5? How many edges are in Kn?

Wouldn't the edges be at certain points of the graph? For instance, Point 1, Point 2, Point 3, Point 4, and Point 5 or n-1, n-2, n-3, n-4, and n-5.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
I don't quite understand what you mean by
Wouldn't the edges be at certain points of the graph? For instance, Point 1, Point 2, Point 3, Point 4, and Point 5 or n-1, n-2, n-3, n-4, and n-5.
An edge in a (simple) graph is an unordered pair of distinct vertices in that graph. Any complete graph on $n$ vertices has ${n\choose 2} = n(n-1)/2$ edges in it. Thus $K_5$ has $10$ edges.

#### Joystar1977

##### Active member
Is this correct then that since the graph would have 5 vertices, denoted by K5 then I would solve for Kn like this:

5 (5 - 1) / 2

5 (4) / 2

20 / 2

10

Are the edges of Kn equal to 10? If not, someone please correct this for me.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Is this correct then that since the graph would have 5 vertices, denoted by K5 then I would solve for Kn like this:

5 (5 - 1) / 2

5 (4) / 2

20 / 2

10

Are the edges of Kn equal to 10? If not, someone please correct this for me.
Yes, there are $10$ edges in $K_5$. Your calculation is correct. I thin you meant to write $K_5$ instead of the thing marked in red in the quoted text.

#### Joystar1977

##### Active member
Caffeine Machine: The question of Part B has two separate questions. Both of them are as follows:

Consider the complete graph with 5 vertices, denoted by K5.

1. How many edges are in K5?

2. How many edges are in Kn?

I was asking whether or not the K5 and Kn have the same amount of edges. I know you said that K5 has ten edges, so does that mean that Kn has the same amount of edges?

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
I was asking whether or not the K5 and Kn have the same amount of edges. I know you said that K5 has ten edges, so does that mean that Kn has the same amount of edges?
Of course not. For example, K2 consists of two connected vertices; it has 1 edge. K3 is a triangle; it has three edges. K4 is a square with both diagonals; it has 6 edges. Post #2 has the formula for an arbitrary $n$, but you may have missed it. Each vertex of Kn is connected to $n - 1$ other vertices, so there are $n - 1$ edges coming out of that vertex. If you multiply that by the number of vertices, you get $n(n-1)$. However, in this way you count every edge twice. (Why?)

#### Joystar1977

##### Active member
Evgeny.Makarov! If you say, then that K2 has 1 edge, K3 has 3 edges, and K4 has 6 edges then what about K1 and K5?

K2= 1 edge
K3= 3 edges
K4= 6 edges

Total edges = 10 edges

You stated that n-1 edges come out of that vertex and to multiply by the number of vertices which is 5 vertices. I would get n (n - 1).

5 (10 -1)
5 (9)
45

Is Kn = 45 edges

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
I also would like to ask you some questions.

If you say, then that K2 has 1 edge, K3 has 3 edges, and K4 has 6 edges then what about K1 and K5?
Do you know the definition of $K_n$ for various $n$? If so, then you should know what $K_1$ is: it's a single vertex. There are no edges because there is nothing to connect to. As for $K_5$, you have a drawing in this post in a parallel thread. Why don't you count the number of edges?

K2= 1 edge
K3= 3 edges
K4= 6 edges
Above, you wrote correctly: "K4 has 6 edges". It does not equal 6 edges. Graphs can be compared only to graphs, and the number of edges to the number of some objects.

You stated that n-1 edges come out of that vertex and to multiply by the number of vertices which is 5 vertices.
Let's recall exactly what I said.
Each vertex of Kn is connected to $n - 1$ other vertices, so there are $n - 1$ edges coming out of that vertex. If you multiply that by the number of vertices, you get $n(n-1)$. However, in this way you count every edge twice.
Why are you saying that the number of vertices is 5? Did I talk about $K_5$? No, I talked about $K_n$ for some unknown number $n$.

I would get n (n - 1).

5 (10 -1)
So, you replace the first $n$ in $n (n - 1)$ by 5, and you replace the second $n$ by 10. Do you think this is right?

Is Kn = 45 edges
This again brings me to the question whether you know what $K_n$ is. There are graphs $K_3$, $K_4$, $K_5$, but there is no such object as $K_n$ for unknown $n$. Strictly speaking, $K_n$ is a function: you give it $n$, it returns you a complete graph on $n$ vertices. E.g., you give it 5, it returns $K_5$.

So, I am asking you: what is the meaning of the phrase "$K_n$ has 45 edges"?

#### Joystar1977

##### Active member
Evgeny.Makarov then is it correct to say that Kn doesn't have anything to connect to since it's a single vertex. Earlier I went off what you said about K4 having six edges so I thought that since it has six edges that it was equal to that. The reason I am saying that the number of vertices is 5 because the problem states the following:

Consider the complete graph with 5 vertices, denoted by K5.

How many edges are in K5?

How many edges are in Kn?

My understanding of this when it says the complete graph means the whole entire graph and I would think that if the graph has 5 vertices, denoted by K5 then Kn to my recollection is the unknown amount of edges in the graph. Sometimes I read too much into things so please let me know if I am or not.

When I solve the problem you said to replace the first n (n - 1) by 5 and the second n by 10 so this is what I was solving the problem:

5 (n-1) - Replaced first n by 5

5 (10-1) - Replaced second n by 10

5 (9)- Doing what is in the parentheses first

45 - Multiplying 5 (9)

My understanding of this when you say "Kn has 45 edges" is that "Kn is equal to having 45 edges". Did I forget to do something else?

KN has N vertices.
Each vertex has degree N - 1.
The sum of all degrees is N(N - 1).

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Evgeny.Makarov then is it correct to say that Kn doesn't have anything to connect to since it's a single vertex. Earlier I went off what you said about K4 having six edges so I thought that since it has six edges that it was equal to that. The reason I am saying that the number of vertices is 5 because the problem states the following:

Consider the complete graph with 5 vertices, denoted by K5.

How many edges are in K5?

How many edges are in Kn?

My understanding of this when it says the complete graph means the whole entire graph and I would think that if the graph has 5 vertices, denoted by K5 then Kn to my recollection is the unknown amount of edges in the graph. Sometimes I read too much into things so please let me know if I am or not.

When I solve the problem you said to replace the first n (n - 1) by 5 and the second n by 10 so this is what I was solving the problem:

5 (n-1) - Replaced first n by 5

5 (10-1) - Replaced second n by 10

5 (9)- Doing what is in the parentheses first

45 - Multiplying 5 (9)

My understanding of this when you say "Kn has 45 edges" is that "Kn is equal to having 45 edges". Did I forget to do something else?

KN has N vertices.
Each vertex has degree N - 1.
The sum of all degrees is N(N - 1).
Hey Joystar.

I think you have misunderstood and confused a lot of graph theoretic notations and concepts. I suggest you go through a process of 'rebooting'. You need to forget everything you know about graphs and start again. Do the following exercises:

1) Define a graph.
2) What is an 'edge' of a graph?
3) Does it matter if we join two vertices with a straight line or a curved line when we represent a graph by drawing it on paper?
4) What is a cycle graph? Give an example.
5) What is a complete graph? Give an example.
6) If we consider the $G$ as the whole of the cycle graph having 4 edges then is $G$ a complete graph?

You can post your answers on this thread or on a different thread if you like. I will be happy to check them and comment.

#### Joystar1977

##### Active member

1. Define a graph.

A graph is a two-dimensional drawing showing a relationship usually between two sets of numbers by means of a line, curve, a series of bars, or other symbols. Usually, the independent variable is shown on the horizontal line (X-Axis) and the dependent variable is shown on the vertical line (Y-Axis). When the axis is perpendicular then is intersects as a point called the origin and is calibrated in the units of the quantities represented. A graph usually has four quadrants representing the positive and negative values of the variables. Most of the time the north-east quadrant is shown on the graph when the negative values do not exist or don’t have interest. A graph can also be defined as a diagram of values, usually shown as lines or bars.

2. What is an ‘edge’ of a graph?

The edge of a graph is the intersection of two planes or faces. The edges can also be called arcs. In a simple graph, the set of vertices (V) and set of unordered pairs of distinct elements of V called edges. Graphs are not all simple. Sometimes a pair of vertices are connected by multiple edge yielding a multigraph. At times the vertices are connected to themselves by an edge called a loop, creating a pseudograph. Edges can also be given a direction creating a directed graph. An edge joins one vertex to another or starts and ends at the same vertex. The edges can have straight or curved lines and is also known as arcs.

3. Does it matter if we join two vertices with a straight line or a curved line when we represent a graph by drawing it on paper?

I do know that a straight line should be drawn with a ruler and not free hand. A curved line should be drawn in a smooth “swoop” through the points to indicate the general shape. A curved graph, you need as many points as possible to make it accurate. If your given a straight line in a graph, then it shows that the two physical quantities that I plotted are “proportional”. If the straight line goes through the origin of the graph, then it indicates that they are “directly proportional”. If you double one quantity the other will double too. Any points that are well away from the line are called anomalies. This may be due to experimental error. From my understanding of this that since I don’t think it really matters whether or not you join two vertices with a straight line or curved line because if I am drawing it on paper then I would make the curved line as accurate as possible to match up with the straight line.

4. What is a cycle graph? Give an example.

A cycle graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. An example of a cycle graphs are as follows: Triangle Graph, Square Graph, Grid Graph, Circular Graph, Block Cycle, Step Cycle, Arrow Cycle, and a Balanced Cycle.

5. What is a complete graph? Give an example.

A complete graph is a graph with N vertices and an edge between every two vertices. There are no loops. Every two vertices share exactly one edge. The symbol KN is used for a complete graph with N vertices. Also, a complete graph is a graph in which each pair of graph vertices is connected by an edge. Some examples of a complete graph are as follows: Line graph, Star graph, Planar graph, Wheel Graph, Odd Graph, Tetrahedral Graph, and Pentatope Graph.

6. If we consider the G as the whole of the cycle graph having 4 edges then is G a complete graph?

I would say yes that G is the whole of the cycle graph having 4 edges and would be considered a complete graph. The reason is because a complete graph has an edge between every two vertices, with no loops, and every two vertices share exactly one edge.

#### caffeinemachine

##### Well-known member
MHB Math Scholar

1. Define a graph.

A graph is a two-dimensional drawing showing a relationship usually between two sets of numbers by means of a line, curve, a series of bars, or other symbols. Usually, the independent variable is shown on the horizontal line (X-Axis) and the dependent variable is shown on the vertical line (Y-Axis). When the axis is perpendicular then is intersects as a point called the origin and is calibrated in the units of the quantities represented. A graph usually has four quadrants representing the positive and negative values of the variables. Most of the time the north-east quadrant is shown on the graph when the negative values do not exist or don’t have interest. A graph can also be defined as a diagram of values, usually shown as lines or bars.

2. What is an ‘edge’ of a graph?

The edge of a graph is the intersection of two planes or faces. The edges can also be called arcs. In a simple graph, the set of vertices (V) and set of unordered pairs of distinct elements of V called edges. Graphs are not all simple. Sometimes a pair of vertices are connected by multiple edge yielding a multigraph. At times the vertices are connected to themselves by an edge called a loop, creating a pseudograph. Edges can also be given a direction creating a directed graph. An edge joins one vertex to another or starts and ends at the same vertex. The edges can have straight or curved lines and is also known as arcs.

3. Does it matter if we join two vertices with a straight line or a curved line when we represent a graph by drawing it on paper?

I do know that a straight line should be drawn with a ruler and not free hand. A curved line should be drawn in a smooth “swoop” through the points to indicate the general shape. A curved graph, you need as many points as possible to make it accurate. If your given a straight line in a graph, then it shows that the two physical quantities that I plotted are “proportional”. If the straight line goes through the origin of the graph, then it indicates that they are “directly proportional”. If you double one quantity the other will double too. Any points that are well away from the line are called anomalies. This may be due to experimental error. From my understanding of this that since I don’t think it really matters whether or not you join two vertices with a straight line or curved line because if I am drawing it on paper then I would make the curved line as accurate as possible to match up with the straight line.

4. What is a cycle graph? Give an example.

A cycle graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. An example of a cycle graphs are as follows: Triangle Graph, Square Graph, Grid Graph, Circular Graph, Block Cycle, Step Cycle, Arrow Cycle, and a Balanced Cycle.

5. What is a complete graph? Give an example.

A complete graph is a graph with N vertices and an edge between every two vertices. There are no loops. Every two vertices share exactly one edge. The symbol KN is used for a complete graph with N vertices. Also, a complete graph is a graph in which each pair of graph vertices is connected by an edge. Some examples of a complete graph are as follows: Line graph, Star graph, Planar graph, Wheel Graph, Odd Graph, Tetrahedral Graph, and Pentatope Graph.

6. If we consider the G as the whole of the cycle graph having 4 edges then is G a complete graph?

I would say yes that G is the whole of the cycle graph having 4 edges and would be considered a complete graph. The reason is because a complete graph has an edge between every two vertices, with no loops, and every two vertices share exactly one edge.
Hello Joystar.

Its good that you answered each question. Before I comment on your answers I need to know something about your background in mathematics. Are you an undergraduate? If yes then are you a sophomore or second year or senior? If not then which grade in High School?

Also, are you comfortable with set theoretic concepts, notations and symbols? Do you fully understand the meanings of 'cartesian product of two sets', 'union and intersection of two sets'?

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
I agree with caffeinemachine that you may have misunderstood some definitions. In this case, we may be talking about different things and not realize it. I recommend going back to definitions and making sure that your understanding is correct.

With regard to the discussion above...

Evgeny.Makarov then is it correct to say that Kn doesn't have anything to connect to since it's a single vertex.
Let's recall what I said in post #8:
There are graphs $K_3$, $K_4$, $K_5$, but there is no such object as $K_n$ for unknown $n$. Strictly speaking, $K_n$ is a function: you give it $n$, it returns you a complete graph on $n$ vertices. E.g., you give it 5, it returns $K_5$
First, note that $K_n$ is not an indivisible name like "pentagon". In consists of two parts: the name of the collection $K$, which probably stands for "complete", and an index $n$, which takes values 1, 2, …. As I said, $K_n$ is a function $K$ that takes an argument $n$; it is more like $K(n)$. It may have been confusing to write Kn in plain text because it looks like an indivisible name. Recall functions, e.g., $f(x)=x^2$: such $f(x)$ is not a single number; for each $x$ is returns its own number.

Yet you continue treating $K_n$ as if it is a concrete graph and keep saying things like it has a single vertex or it has 45 edges. $K_n$ is not a single graph, it is a whole collection of graphs: $K_1$, $K_2$, $K_3$ and so on. What is true for $K_1$ (it consists of a single vertex) is not true for $K_4$. What is true for $K_5$ (it has 10 edges) is not true for $K_3$. You cannot say "$K_n$ has 10 edges" and stop there because this is neither true nor false. It may be true for some graphs in the $K_n$ family and false for others. In contrast, the phrase "$K_5$ has 10 edges" has a definite truth value (it is true).

Another way to refer to $K_n$ meaningfully is to use phrases "for all $n$" or "for some $n$". Then you are talking about the collection $K_n$ as a whole. For example, it makes sense to say, "For all $n$, the graph $K_n$ has $n(n-1)/2$ edges". In this one sentence you make an infinite number of claims: $K_1$ has 0 edges, $K_2$ has 1 edge, $K_3$ has 3 edges, $K_4$ has 6 edges and so on. It also makes sense to say, "$K_n$ has 10 edges for some $n$", namely for $n=5$.

Usually when we refer to $K_n$ without specifying concrete $n$ and without using either of the phrases "for all $n$" or "for some $n$" (or their equivalents), then we implicitly mean "for all $n$". Thus, saying "$K_n$ has $n$ vertices" is equivalent to saying "$K_n$ has $n$ vertices for all $n$". Again, this is a claim about the whole collection $K_n$ for all $n$. Thus, "$K_n$ has 45 edges" is false just because there are elements of the collection with a differemt number of edges: e.g., $K_2$ has just 1 edge.

My understanding of this when it says the complete graph means the whole entire graph and I would think that if the graph has 5 vertices, denoted by K5 then Kn to my recollection is the unknown amount of edges in the graph.
"Complete" here is not used in its usual disctionary sense. A "complete graph" is an idivisible name, and it is given a new precise meaning by a definition. Namely, let $n$ be some natural number. Then a complete graph on $n$ vertices, denoted by $K_n$, consists of $n$ vertices, and every pair of distinct vertices is connected by an edge. This is also a whole series of definitions: for $n=1$, $n=2$ and so on. So, the word "complete" does not mean that some prevously defined graph is considered in its entirety; it is a part of a name that is given a new meaning. (Redefining familiar terms happens a lot in mathematics: we talk of "compact" and "meager" sets, properties holding "almost everywhere" or "eventually" and so on. Definitions give these terms new precise meanings different from those in the dictionary.) Similarly, $K_n$ is not a graph with an unknown number of edges. You can't talk about the number of edges in $K_n$ until you either give a concrete $n$ or use the phrases "for all $n$" or "for some $n$".

When I solve the problem you said to replace the first n (n - 1) by 5 and the second n by 10
Let's recall exactly:
So, you replace the first $n$ in $n (n - 1)$ by 5, and you replace the second $n$ by 10. Do you think this is right?
I did not direct you to replace $n$ by two different numbers. I stated this as an observation and asked if you think it right. Have you ever seen the same variable in a single expression replaced by two values? I was hoping you notice this and feel ashamed because this is not done in mathematics.

My understanding of this when you say "Kn has 45 edges" is that "Kn is equal to having 45 edges".
In addition to the above, I want to stress that having 45 edges is not the same as being equal to 45 edges. Presumably, you have a computer, but you are not equal to a computer, right? I am not sure if the phrase "equal to having" is possible in English, but it is not correct here. "Having 45 edges" is at best a property of graphs, and a graph $K_n$ cannot be equal to a property, or a number, or a function and so on. A graph may only be equal to a graph. Theerefore, please don't say, "A graph is equal to 45 edges" or "A graph is equal to having 45 edges"; this is absolutely wrong.

- - - Updated - - -

1. Define a graph.

A graph is a two-dimensional drawing showing a relationship usually between two sets of numbers by means of a line, curve, a series of bars, or other symbols. Usually, the independent variable is shown on the horizontal line (X-Axis) and the dependent variable is shown on the vertical line (Y-Axis).
No. Just no. Not in the context of this thread..

#### Joystar1977

##### Active member
Caffeinemachine:

In response to your questions, I am a third year undergraduate (Junior) in college and have taken Basic Mathematics, Review of Arithmetic, Elementary Algebra, Intermediate Algebra, and am now presently taking Discrete Mathematics. No, I am not very comfortable with the set theoretic concepts, notations and symbols. No, I don't fully understand the meanings of a "cartesian product of two sets" and "union and intersection of two sets".

Hello Joystar.

Its good that you answered each question. Before I comment on your answers I need to know something about your background in mathematics. Are you an undergraduate? If yes then are you a sophomore or second year or senior? If not then which grade in High School?

Also, are you comfortable with set theoretic concepts, notations and symbols? Do you fully understand the meanings of 'cartesian product of two sets', 'union and intersection of two sets'?

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Caffeinemachine:

In response to your questions, I am a third year undergraduate (Junior) in college and have taken Basic Mathematics, Review of Arithmetic, Elementary Algebra, Intermediate Algebra, and am now presently taking Discrete Mathematics. No, I am not very comfortable with the set theoretic concepts, notations and symbols. No, I don't fully understand the meanings of a "cartesian product of two sets" and "union and intersection of two sets".
Hmm..

To understand the concepts of Graph Theory thoroughly you first need to understand some basic set theoretic concepts and get comfortable with the notations.

A 'cartesian product' of two sets $A$ and $B$ is written as $A\times B$ and is defined as $A\times B=\{(a,b):a\in A, b\in B\}$. That is, the cartesian product of $A$ and $B$ is the 'collection' of all 'ordered pairs' which have an element of $A$ in the first entry and an element of $B$ in the second entry. This is a fairly intuitive concept.

Union of two sets $A$ and $B$ is written as $A\cup B$ and is the collection of all elements of $A$ and $B$. There may be common elements in $A$ and $B$. That's fine. Don't let it bother you.

Intersection of $A$ and $B$ is written as $A\cap B$ and is the collection of all the elements which are found in both $A$ and $B$.

Now,
Your definition of a Graph is not correct. We like to think of graphs as diagrams on paper but we cannot define a graph as a diagram on paper. The definition is:
A graph $G$ is an ordered pair of a vertex set $V$ and an edge set $E$, where $E$ is a collection of some two element subsets of $V$. We write $G=(V,E)$.

The above definition has nothing to do with a diagram. It's quite abstract. To visualize a graph we represent the vertices as dots and the edges as lines(curved or straight or zig zag or dotted or whatever) connecting two distinct vertices.

Tell me if you have further doubts in any of these. Once we are through with the definition of a graph I can comment on other answers of yours.

#### Joystar1977

##### Active member
Caffeinemachine: Just so you know there is no dictionary in the back of my mathematics textbook, so these answers I got was from online of a graph theory Glossary. Of course, now that I have written this down go ahead and please explain further about the other questions I answered.

#### Joystar1977

##### Active member
Evgeny.Makarov: I may have misunderstood some definitions, but as I stated below to Caffeinemachine there is no dictionary in the back of my textbook so the answers I got was from online of a Graph Theory Glossary. I am not trying to talk about different things here, but what I am trying to figure out is what is Kn. I will tell you right now I never took any upper division math courses in high school and was in Special Education all of my life. I have a learning disability (Epilepsy Seizures) which causes me to have a hard time comprehending and understanding. Also, just so you know that I have repeated Elementary Algebra a total of 4 times and Intermediate Algebra a total of 5 times. As for in response to your question, I am not going to feel ashamed just because I didn't notice or observe something. Furthermore, I might add that no I haven't seen the same variable used in a single expression or equation that is replaced by two values. Finally, I will not say that "having 45 edges" is "equal to having 45 edges".

I agree with caffeinemachine that you may have misunderstood some definitions. In this case, we may be talking about different things and not realize it. I recommend going back to definitions and making sure that your understanding is correct.

With regard to the discussion above...

Let's recall what I said in post #8:First, note that $K_n$ is not an indivisible name like "pentagon". In consists of two parts: the name of the collection $K$, which probably stands for "complete", and an index $n$, which takes values 1, 2, …. As I said, $K_n$ is a function $K$ that takes an argument $n$; it is more like $K(n)$. It may have been confusing to write Kn in plain text because it looks like an indivisible name. Recall functions, e.g., $f(x)=x^2$: such $f(x)$ is not a single number; for each $x$ is returns its own number.

Yet you continue treating $K_n$ as if it is a concrete graph and keep saying things like it has a single vertex or it has 45 edges. $K_n$ is not a single graph, it is a whole collection of graphs: $K_1$, $K_2$, $K_3$ and so on. What is true for $K_1$ (it consists of a single vertex) is not true for $K_4$. What is true for $K_5$ (it has 10 edges) is not true for $K_3$. You cannot say "$K_n$ has 10 edges" and stop there because this is neither true nor false. It may be true for some graphs in the $K_n$ family and false for others. In contrast, the phrase "$K_5$ has 10 edges" has a definite truth value (it is true).

Another way to refer to $K_n$ meaningfully is to use phrases "for all $n$" or "for some $n$". Then you are talking about the collection $K_n$ as a whole. For example, it makes sense to say, "For all $n$, the graph $K_n$ has $n(n-1)/2$ edges". In this one sentence you make an infinite number of claims: $K_1$ has 0 edges, $K_2$ has 1 edge, $K_3$ has 3 edges, $K_4$ has 6 edges and so on. It also makes sense to say, "$K_n$ has 10 edges for some $n$", namely for $n=5$.

Usually when we refer to $K_n$ without specifying concrete $n$ and without using either of the phrases "for all $n$" or "for some $n$" (or their equivalents), then we implicitly mean "for all $n$". Thus, saying "$K_n$ has $n$ vertices" is equivalent to saying "$K_n$ has $n$ vertices for all $n$". Again, this is a claim about the whole collection $K_n$ for all $n$. Thus, "$K_n$ has 45 edges" is false just because there are elements of the collection with a differemt number of edges: e.g., $K_2$ has just 1 edge.

"Complete" here is not used in its usual disctionary sense. A "complete graph" is an idivisible name, and it is given a new precise meaning by a definition. Namely, let $n$ be some natural number. Then a complete graph on $n$ vertices, denoted by $K_n$, consists of $n$ vertices, and every pair of distinct vertices is connected by an edge. This is also a whole series of definitions: for $n=1$, $n=2$ and so on. So, the word "complete" does not mean that some prevously defined graph is considered in its entirety; it is a part of a name that is given a new meaning. (Redefining familiar terms happens a lot in mathematics: we talk of "compact" and "meager" sets, properties holding "almost everywhere" or "eventually" and so on. Definitions give these terms new precise meanings different from those in the dictionary.) Similarly, $K_n$ is not a graph with an unknown number of edges. You can't talk about the number of edges in $K_n$ until you either give a concrete $n$ or use the phrases "for all $n$" or "for some $n$".

Let's recall exactly:
I did not direct you to replace $n$ by two different numbers. I stated this as an observation and asked if you think it right. Have you ever seen the same variable in a single expression replaced by two values? I was hoping you notice this and feel ashamed because this is not done in mathematics.

In addition to the above, I want to stress that having 45 edges is not the same as being equal to 45 edges. Presumably, you have a computer, but you are not equal to a computer, right? I am not sure if the phrase "equal to having" is possible in English, but it is not correct here. "Having 45 edges" is at best a property of graphs, and a graph $K_n$ cannot be equal to a property, or a number, or a function and so on. A graph may only be equal to a graph. Theerefore, please don't say, "A graph is equal to 45 edges" or "A graph is equal to having 45 edges"; this is absolutely wrong.

- - - Updated - - -

No. Just no. Not in the context of this thread..

#### caffeinemachine

##### Well-known member
MHB Math Scholar

1. Define a graph.

A graph is a two-dimensional drawing showing a relationship usually between two sets of numbers by means of a line, curve, a series of bars, or other symbols. Usually, the independent variable is shown on the horizontal line (X-Axis) and the dependent variable is shown on the vertical line (Y-Axis). When the axis is perpendicular then is intersects as a point called the origin and is calibrated in the units of the quantities represented. A graph usually has four quadrants representing the positive and negative values of the variables. Most of the time the north-east quadrant is shown on the graph when the negative values do not exist or don’t have interest. A graph can also be defined as a diagram of values, usually shown as lines or bars.
This is completely different meaning of a graph. This usage of the term graph applies to the sketch of functions. Like when you want to picturize $f(x)=x^2$ for real $x$. We are not using the word graph to mean this at all. Here we use the word graph as used in the context of discrete mathematics. Here is a link Graph theory - Wikipedia, the free encyclopedia
I have defined graphs in a previous post already.

2. What is an ‘edge’ of a graph?

The edge of a graph is the intersection of two planes or faces. The edges can also be called arcs. In a simple graph, the set of vertices (V) and set of unordered pairs of distinct elements of V called edges. Graphs are not all simple. Sometimes a pair of vertices are connected by multiple edge yielding a multigraph. At times the vertices are connected to themselves by an edge called a loop, creating a pseudograph. Edges can also be given a direction creating a directed graph. An edge joins one vertex to another or starts and ends at the same vertex. The edges can have straight or curved lines and is also known as arcs.
The thing marked in red is an incorrect statement. That definition applies to a particular class of graphs, called planar graphs, in a special sense and we can dispense with that for now. The right answer is that an edge of a graph $G$ is simply any element in the edge set $E(G)$ of the graph $G$.

3. Does it matter if we join two vertices with a straight line or a curved line when we represent a graph by drawing it on paper?

I do know that a straight line should be drawn with a ruler and not free hand. A curved line should be drawn in a smooth “swoop” through the points to indicate the general shape. A curved graph, you need as many points as possible to make it accurate. If your given a straight line in a graph, then it shows that the two physical quantities that I plotted are “proportional”. If the straight line goes through the origin of the graph, then it indicates that they are “directly proportional”. If you double one quantity the other will double too. Any points that are well away from the line are called anomalies. This may be due to experimental error. From my understanding of this that since I don’t think it really matters whether or not you join two vertices with a straight line or curved line because if I am drawing it on paper then I would make the curved line as accurate as possible to match up with the straight line.
Again, here you probably had the other usage of the word graph in mind while writing this answer.
The right answer is that it does not matter how you draw an edge to connect two vetices.

4. What is a cycle graph? Give an example.

A cycle graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. An example of a cycle graphs are as follows: Triangle Graph, Square Graph, Grid Graph, Circular Graph, Block Cycle, Step Cycle, Arrow Cycle, and a Balanced Cycle.
The things marked in blue are probably not cycle graphs. For a definition see Cycle (graph theory) - Wikipedia, the free encyclopedia

5. What is a complete graph? Give an example.

A complete graph is a graph with N vertices and an edge between every two vertices. There are no loops. Every two vertices share exactly one edge. The symbol KN is used for a complete graph with N vertices. Also, a complete graph is a graph in which each pair of graph vertices is connected by an edge. Some examples of a complete graph are as follows: Line graph, Star graph, Planar graph, Wheel Graph, Odd Graph, Tetrahedral Graph, and Pentatope Graph.
The definition of a complete graph looks okay. But Many of the examples (marked in blue) are not complete graphs. Find out which ones.

6. If we consider the G as the whole of the cycle graph having 4 edges then is G a complete graph?

I would say yes that G is the whole of the cycle graph having 4 edges and would be considered a complete graph. The reason is because a complete graph has an edge between every two vertices, with no loops, and every two vertices share exactly one edge.
No, a cycle having 4 edges is not a complete graph. This follows easily from the definitions.

#### Jameson

Staff member
Evgeny.Makarov: I may have misunderstood some definitions, but as I stated below to Caffeinemachine there is no dictionary in the back of my textbook so the answers I got was from online of a Graph Theory Glossary. I am not trying to talk about different things here, but what I am trying to figure out is what is Kn. I will tell you right now I never took any upper division math courses in high school and was in Special Education all of my life. I have a learning disability (Epilepsy Seizures) which causes me to have a hard time comprehending and understanding. Also, just so you know that I have repeated Elementary Algebra a total of 4 times and Intermediate Algebra a total of 5 times. As for in response to your question, I am not going to feel ashamed just because I didn't notice or observe something. Furthermore, I might add that no I haven't seen the same variable used in a single expression or equation that is replaced by two values. Finally, I will not say that "having 45 edges" is "equal to having 45 edges".
Hi Joystar1977,

I've been following this thread with interest since I am completely ignorant on the topic. I've been part of math sites for quite some time now and over the years I've noticed that mathematicians have a tendency to be straight to the point and don't sugar coat things, but that doesn't always mean they are trying to be rude or hurt your feelings. Both of the members in this thread are trying to help, trust me. They are both "MHB Math Helpers" and being respectful to other members is a huge part of that title.

Of course you shouldn't feel ashamed!! You are working hard to further your knowledge of a difficult topic. You're in the right place here at MHB at I hope that you feel we are here to help you.

Jameson

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Hi Joystar1977,

I've been following this thread with interest since I am completely ignorant on the topic. I've been part of math sites for quite some time now and over the years I've noticed that mathematicians have a tendency to be straight to the point and don't sugar coat things, but that doesn't always mean they are trying to be rude or hurt your feelings. Both of the members in this thread are trying to help, trust me. They are both "MHB Math Helpers" and being respectful to other members is a huge part of that title.

Of course you shouldn't feel ashamed!! You are working hard to further your knowledge of a difficult topic. You're in the right place here at MHB at I hope that you feel we are here to help you.

Jameson
Thank you Jameson for these encouraging lines for Joystar.
I couldn't have said it better myself.
___

You are doing a good job Joystar. Just keep trying and you will have these concepts down in no time.

#### Joystar1977

##### Active member
Jameson: I feel the same way in the sense that if I knew about this course material, then I wouldn't be here seeking help. I thought that it would help me to tell Evgeny.Makarov and Caffeinemachine a little about myself because of the fact they wouldn't get so sick and tired of me for not understanding the material or kind of saying to themselves, "How come this person isn't understanding this material?" Everybody has their strengths and weaknesses and I admit mine is not mathematics. I am really good at writing, English, Spelling, Grammar, Punctuation, Vocabulary, Reading, and Public Speaking. I wasn't trying to be rude and sorry if you thought I was but I am the type of person, "who takes things for what they mean" and also have the belief "that if a person didn't mean to say something, then they wouldn't popped their mouth off in the first place." It just basically means to think before you speak and all I ask is for them to have patience with me because I do have a learning disability (Epilepsy Seizures) while I try to get through this course (Discrete Mathematics). This is very difficult and it should tell you a lot about myself especially after I repeated Elementary Algebra 4 times and Intermediate Algebra 5 times and finally ended up passing them. I know that the MHB Math Helpers are here to help me. I am learning as I go through this and try to grasp these concepts.

Hi Joystar1977,

I've been following this thread with interest since I am completely ignorant on the topic. I've been part of math sites for quite some time now and over the years I've noticed that mathematicians have a tendency to be straight to the point and don't sugar coat things, but that doesn't always mean they are trying to be rude or hurt your feelings. Both of the members in this thread are trying to help, trust me. They are both "MHB Math Helpers" and being respectful to other members is a huge part of that title.

Of course you shouldn't feel ashamed!! You are working hard to further your knowledge of a difficult topic. You're in the right place here at MHB at I hope that you feel we are here to help you.

Jameson

#### Joystar1977

##### Active member
Caffeinemachine: Thanks for these encouraging words and I am on an extension for my course at the college I am presently attending. I guess then not everything you read online (referring to the Internet) is correct, because as I mentioned to you before I got my answers from a Graph Theory Glossary here on the Internet. Please let's get back to what we were talking about.

I know that a complete graph is a graph (means one) with N vertices (means more than one) and an edge (means one) between every two vertices (means more than one). There are no loops (means more than one) in a complete graph. Every two vertices (means more than one) share exactly one edge (means one). Also, a complete graph (means one) is a graph (means one) in which each pair (means one) of graph vertices (means more than one) is connected by an edge (means one).

Please explain further about how come G is not considered the whole of the cycle graph when it has 4 edges so its not considered a complete graph. My understanding of the cycle graph is that it consists of a single cycle (to me this means one) or in other words specifying some number (to me this means any number) of vertices (to me this means more than one) connected in a closed chain (to me this means that in order for a chain to be closed and connected then it would have to have more than one connection or vertice). Am I reading too much into this?

Thank you Jameson for these encouraging lines for Joystar.
I couldn't have said it better myself.
___

You are doing a good job Joystar. Just keep trying and you will have these concepts down in no time.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
I know that a complete graph is a graph (means one) with N vertices (means more than one) and an edge (means one) between every two vertices (means more than one). There are no loops (means more than one) in a complete graph. Every two vertices (means more than one) share exactly one edge (means one). Also, a complete graph (means one) is a graph (means one) in which each pair (means one) of graph vertices (means more than one) is connected by an edge (means one).
Good job. Just one thing. The thing marked in blue is redundant. You have already stated in the black text what you have said in the blue text. Feel free to ask me if this confuses you.

Please explain further about how come G is not considered the whole of the cycle graph when it has 4 edges so its not considered a complete graph. My understanding of the cycle graph is that it consists of a single cycle (to me this means one) or in other words specifying some number (to me this means any number) of vertices (to me this means more than one) connected in a closed chain (to me this means that in order for a chain to be closed and connected then it would have to have more than one connection or vertice). Am I reading too much into this?
Hmm..
Have a look at this https://docs.google.com/file/d/0B77QF0wgZJZ7NWZpbWQzakVybW8/edit
The left diagram is that of a cycle graph on 4 vetrices and the right diagram is that of a complete graph on 4 vertices. Note that in the left diagram the vertices $a$ and $d$ are not connected. Thus the left diagram cannot be a diagram of a complete graph as any two distinct vertices in a complete graph have an edge joining them.

I'd like to further your understanding of a cycle graph. First we need the concept of a cycle in any graph. Let $G=(V,E)$ be any graph. A cycle in $G$ is a sequence of vertices $v_1,\ldots,v_k$ such that
1. $v_i\neq v_j$ if $i\neq j$
2. $v_i$ and $v_{i+1}$ have an edge between them for all $1\leq i\leq k-1$
3. $v_1$ and $v_k$ have an edge between them.
The concept of a cycle is quite intuitive, as are most of the concepts in graph theory.

Now what is a cycle graph? A cycle graph is graph $G$ such that there is a sequence of vertices of $G$ such that
1. This sequence has all the vertices of $G$
2. This sequence forms a cycle in $G$.
3. There are no other edges in $G$ other than those found in the cycle formed by this sequence.

This may sound too technical and hard to understand. But the trick is to first see the intuitive picture and then try to write it formally on paper. This just looks complicated. Believe me, it's nothing like what it sounds.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
I thought that it would help me to tell Evgeny.Makarov and Caffeinemachine a little about myself because of the fact they wouldn't get so sick and tired of me for not understanding the material or kind of saying to themselves, "How come this person isn't understanding this material?"

Somehow the world of mathematics has generated this notion that if a person $X$ is having difficulty in understanding a mathematical concept $C$, then a person $Y$, who understands $C$ properly, will look down on person $X$. It is true that this sometimes happens. But as far as I understand, a real mathematician, an artist, knows that different people have different way of looking at things. Some people prefer a very formal description of things (like me). Something like "Let $x\in A\cap B$" is preferred to "Let $x$ be an element of both $A$ and $B$". And the cartesian product of a family of sets $X_\alpha$ indexed by the set $J$ would written as $\prod_{\alpha\in J}X_\alpha$ in a manner which would emulate passing a death sentence on someone. I believe The Bourbaki were of this category.

But there are also mathematicians who don't like expressing things with so much affectation. They'd use words similar to that of everyday language. I think V. Arnold would come under this class.

When I started reading math I was terrible at things. In the beginning the concept of a function, and concepts related to a function like inverse etc., would elude me beyond anything. The real crux of the concept gets lost in the words use to express it. But with time you start getting a hang of it and one day realize 'It was that simple.. damn' with a smile on your face.

#### Joystar1977

##### Active member
Caffeinemachine: That is a lot easier to understand when you say the fact of what a cycle graph is and I guess from my understanding not all graph theory glossaries are correct or everybody has their own style of teaching. As I have said there are different types of learners and for me I am a hands on to where I have to do mostly everything by hand trying it out not so much writing. As you said that it was redundant, now I kind of know that I was reading too much into things. I believe you that it is nothing like what it sounds. Alright I took a look at the diagram. What is next?

Now what is a cycle graph? A cycle graph is graph $G$ such that there is a sequence of vertices of $G$ such that
1. This sequence has all the vertices of $G$
2. This sequence forms a cycle in $G$.
3. There are no other edges in $G$ other than those found in the cycle formed by this sequence.
Good job. Just one thing. The thing marked in blue is redundant. You have already stated in the black text what you have said in the blue text. Feel free to ask me if this confuses you.

Hmm..
Have a look at this https://docs.google.com/file/d/0B77QF0wgZJZ7NWZpbWQzakVybW8/edit
The left diagram is that of a cycle graph on 4 vetrices and the right diagram is that of a complete graph on 4 vertices. Note that in the left diagram the vertices $a$ and $d$ are not connected. Thus the left diagram cannot be a diagram of a complete graph as any two distinct vertices in a complete graph have an edge joining them.

I'd like to further your understanding of a cycle graph. First we need the concept of a cycle in any graph. Let $G=(V,E)$ be any graph. A cycle in $G$ is a sequence of vertices $v_1,\ldots,v_k$ such that
1. $v_i\neq v_j$ if $i\neq j$
2. $v_i$ and $v_{i+1}$ have an edge between them for all $1\leq i\leq k-1$
3. $v_1$ and $v_k$ have an edge between them.
The concept of a cycle is quite intuitive, as are most of the concepts in graph theory.

Now what is a cycle graph? A cycle graph is graph $G$ such that there is a sequence of vertices of $G$ such that
1. This sequence has all the vertices of $G$
2. This sequence forms a cycle in $G$.
3. There are no other edges in $G$ other than those found in the cycle formed by this sequence.

This may sound too technical and hard to understand. But the trick is to first see the intuitive picture and then try to write it formally on paper. This just looks complicated. Believe me, it's nothing like what it sounds.


Suppose $I$ is a set, called the index set, and with each $i\in I$ we associate a set $A_i$. We call $\{A_i:i\in I\}$ an indexed family of sets. Sometimes this is denoted by $\{A_i\}_{i\in I}$.

Example 1.6.1 Suppose $I$ is the days of the year, and for each $i\in I$, $A_i$ is the set of people whose birthday is $i$, so, for example, $\hbox{Beethoven}\in A_{({\scriptsize\hbox{December 16}})}$. $\square$

Example 1.6.2 Suppose $I$ is the integers and for each $i\in I$, $A_i$ is the set of multiples of $i$, that is, $A_i=\{x\in \Z: i\vert x\}$ (recall that $i|x$ means that $x$ is a multiple of $i$; it is read "$i$ divides $x$''). $\square$

Given an indexed family $\{A_i:i\in I\}$ we can define the intersection and union of the sets $A_i$ using the universal and existential quantifiers: \eqalign{\bigcap_{i\in I} A_i & = \{ x: \forall i\in I\,(x\in A_i)\}\cr \bigcup_{i\in I} A_i & = \{ x: \exists i\in I\,(x\in A_i)\}.\cr}

Example 1.6.3 If $\{A_i:i\in I\}$ is the indexed family of example 1.6.1, then $\bigcap_{i\in I} A_i$ is the empty set and $\bigcup_{i\in I} A_i$ is the set of all people. If $\{A_i:i\in I\}$ is the indexed family of example 1.6.2, then $\bigcap_{i\in I} A_i$ is $\{0\}$ and $\bigcup_{i\in I} A_i$ is the set of all integers. $\square$

Since the intersection and union of an indexed family are essentially "translations'' of the universal and existential quantifiers, it should not be too surprising that there are De Morgan's laws that apply to these unions and intersections.

Theorem 1.6.4 If $\{A_i:i\in I\}$ is an indexed family of sets then

a) $(\bigcap_{i\in I} A_i)^c = \bigcup_{i\in I} A_i^c$,

b) $(\bigcup_{i\in I} A_i)^c = \bigcap_{i\in I} A_i^c$.

Proof. We'll do (a): $x\in (\bigcap_{i\in I} A_i)^c$ iff $\lnot (x\in \bigcap_{i\in I} A_i)$ iff $\lnot \forall i\,{\in}\, I\,(x\in A_i)$ iff $\exists i\,{\in}\, I\,(x\notin A_i)$ iff $\exists i\,{\in}\, I\,(x\in A_i^c)$ iff $x\in \bigcup_{i\in I} A_i^c$. $\qed$

You may be puzzled by the inclusion of this theorem: is it not simply part of theorem 1.5.6? No: theorem

Theorem 1.6.5 If $\{A_i:i\in I\}$ is an indexed family of sets and $B$ is any set, then

Proof. Part (a) is a case of specialization, that is, if $x\in \bigcap_{i\in I} A_i$, then $x\in A_i$ for all $i\in I$, in particular, when $i=j$. Part (d) also is easy—if $x\in \bigcup_{i\in I} A_i$, then for some $i\in I$, $x\in A_i\subseteq B$, so $x\in B$. Parts (b) and (c) are left as exercises. $\qed$

An indexed family $\{A_i:i\in I\}$ is pair-wise disjoint if $A_i\cap A_j=\emptyset$ whenever $i$ and $j$ are distinct elements of $I$. The indexed family of example 1.6.1 is pair-wise disjoint, but the one in example 1.6.2 is not. If $S$ is a set then an indexed family $\{A_i:i\in I\}$ of subsets of $S$ is a partition of $S$ if it is pair-wise disjoint and $S= \bigcup_{i\in I} A_i$. Partitions appear frequently in mathematics.

Example 1.6.6 Let $I=\{e,o\}$, $A_e$ be the set of even integers and $A_o$ be the set of odd integers. Then $\{A_i:i\in I\}$ is a partition of $S=\Z$. $\square$

Example 1.6.7 Let $I=\R$, $S=\R^2$, and for each $i\in I$, let $A_i=\{(x,i):x\in \R\}$. Each $A_i$ is a horizontal line and the indexed family partitions the plane. $\square$

Sometimes we want to discuss a collection of sets (that is, a set of sets) even though there is no natural index present. In this case we can use the collection itself as the index.

Example 1.6.8 If ${\cal S}=\{\{1,3,4\}, \{2,3,4,6\}, \{3,4,5,7\}\}$, then $\bigcap_{A\in {\cal S}}A =\{3,4\}$ and $\bigcup_{A\in {\cal S}}A =\{1,2,3,4,5,6,7\}$. $\square$

An especially useful collection of sets is the power set of a set: If $X$ is any set, the power set of $X$ is ${\cal P}(X)=\{A:A\subseteq X\}$.

Example 1.6.9 If $X=\{1,2\}$, then ${\cal P}(X)=\{\emptyset,\{1\},\{2\},\{1,2\}\}$. $\square$

Example 1.6.10 ${\cal P}(\emptyset)=\{\emptyset\}$, that is, the power set of the empty set is non-empty. $\square$

## Exercises 1.6

Ex 1.6.1 Let $I=\{1,2,3\}$, $A_1=\{1,3,4,6,7\}$ $A_2=\{1,4,5,7,8,9\}$, $A_3= \{2,4,7,10\}$. Find $\bigcap_{i\in I} A_i$ and $\bigcup_{i\in I} A_i$.

Ex 1.6.2 Suppose $I=[0,1]\subseteq \R$ and for each $i\in I$, let $A_i=(i-1,i+1)\subseteq \R$. Find $\bigcap_{i\in I} A_i$ and $\bigcup_{i\in I} A_i$.

Ex 1.6.3 Prove parts (b) and (c) of theorem 1.6.5.

Ex 1.6.4 Suppose $U$ is the universe of discourse and the index set $I=\emptyset$. What should we mean by $\bigcap_{i\in I} A_i$ and $\bigcup_{i\in I} A_i$? Show that Theorem 1.6.4 still holds using your definitions.

Ex 1.6.5 Suppose $\{A_i\}_{i\in I}$ is a partition of a set $S$. If $T\subseteq S$, show that $\{A_i\cap T\}_{i\in I}$ is a partition of $T$.

Ex 1.6.6 A collection of sets, $\cal S$, is totally ordered if for every $A,B\in {\cal S}$, either $A\subseteq B$ or $B\subseteq A$. If $\cal S$ is totally ordered, show that ${\cal S}^c=\{A^c: A\in {\cal S}\}$ is totally ordered.

Ex 1.6.7 Suppose $\cal S$ is a collection of sets and $B$ is some other set. Show that if $B$ is disjoint from every $A\in {\cal S}$ then $B$ is disjoint from $\bigcup_{A\in {\cal S}} A$.