1. Chapter 2
Sets and Functions

2. Section 2.2
Relations

3. In listing the elements of a set, the order is not important.
When we wish to indicate that a set of two elements a and b is ordered, we enclose the
elements in parentheses: (a, b).
Then a is called the first element and b is called the second.
The important property of ordered pairs is that
(a, b) = (c, d ) iff a = c and b = d.
Essentially, we
identify the two elements in the ordered pair and specify which one comes first.
Ordered pairs can be defined using basic set theory in a clever way.
Definition 2.2.1
In symbols
we have (a, b) = {{a}, {a, b}}, where we have written the singleton set first.
The ordered pair (a, b) is the set whose members are {a, b} and {a}.
The acceptability of this definition depends on the ordered pairs actually having the
property expected of them.
This we prove in the following theorem.

4. Definition: (a, b) = {{a}, {a, b}} and (c, d) = {{c}, {c, d}}
Theorem 2.2.2: (a, b) = (c, d) iff a = c and b = d.
Proof: If a = c and b = d, then
(a, b) = {{a}, {a, b}} = {{c}, {c, d}} = (c, d).
Then we have {{a}, {a, b}} = {{c}, {c, d}}.
Conversely, suppose that (a, b) = (c, d).
We wish to conclude that
a = c and b = d.
Consider two cases: when a = b and when a  b.
(a, b) = {{a}}.
If a = b, then {a} = {a, b}, so
Since (a, b) = (c, d), we then have
{{a}} = {{c}, {c, d}}.
The set on the left has only one member, {a}.
Thus the set on the right can have only one member, so
But then {{a}} = {{c}}, so
{c} = {c, d} and c = d.
{a} = {c} and a = c.
Thus, a = b = c = d.
On the other hand,
if a  b …

5. Definition: (a, b) = {{a}, {a, b}} and (c, d) = {{c}, {c, d}}
Theorem 2.2.2: (a, b) = (c, d) iff a = c and b = d.
On the other hand, if a  b,
then from the preceding argument it follows that c  d.
Since (a, b) = (c, d), we must have
{a}  {{c}, {c, d}},
which means that {a} = {c} or {a} = {c, d}.
In either case, we have c  {a}, so
a = c.
Again, since (a, b) = (c, d), we must have
{a, b}  {{c}, {c, d}}.
Thus {a, b} = {c} or {a, b} = {c, d}.
But {a, b} has two distinct members and {c} has only one, so we must have
{a, b} = {c, d}.
Now a = c, a  b and b  {c, d}, which implies that
b = d. 

6. Definition 2.2.4
If A and B are sets, then the Cartesian product (or cross product) of A and B,
written A  B, is the set of all ordered pairs (a, b) such that a  A and b  B.
In symbols, A  B = {(a, b) : a  A and b  B}.
Example 2.2.5
If A and B are intervals of real numbers, then in the Cartesian coordinate system with
A on the horizontal axis and B on the vertical axis, A  B is represented by a rectangle.
For example, if A is the interval [1, 4)
then A  B is the
rectangle shown below:
and B is the interval (2, 4], 1 4 2 x y
Note: The solid lines indicate the
left and top edges are included.
[ )
The dashed lines indicate the right
and bottom edges are not included. B
A  B A
[ )

7. A relation between two objects a and b is a condition involving a and b that is either
true or false.
When it is true, we say a is related to b; otherwise, a is not related to b.
For example, “less than” is a relation between positive integers.
We have < 3 is true, < 7 is true, < 4 is false.
When considering a relation between two objects, it is necessary to know which object
comes first.
So it is natural for the formal
definition of a relation to depend on the concept of an ordered pair.
For instance, 1 < 3 is true but 3 < 1 is false.
Definition 2.2.7
Let A and B be sets. A relation between A and B is any subset R of A  B.
We say that an element a in A is related by R to an element b in B if (a, b)  R,
and we often denote this by writing “a R b.”
The first set A is referred to as the domain of the relation and denoted by dom R.
If B = A, then we speak of a relation R  A  A being a relation on A.

8. Example 2.2.8
Returning to Example where A = [1, 4) and B = (2, 4], the relation a R b given by
“a < b” is graphed as the portion of A  B that lies to the left of the line x = y. 1 4 2 x y
[ )
[ ) A B
A  B
x = y R
x = 1
x = 3
For example, we can see that 1 in A is related to all b in (2, 4].
But 3 in A is related only to those b that are in (3, 4].

9. Certain relations are singled out because they possess the properties naturally associated
with the idea of equality.
Definition 2.2.9
A relation R on a set S is an equivalence relation if it has the following properties
for all x, y, z in S:
(a) x R x (reflexive property)
(b) If x R y, then y R x. (symmetric property)
(c) If x R y and y R z, then x R z. (transitive property)
Example
Determine which properties apply to each relation.
(a) Define a relation R on by x R y if x  y.
It is reflexive and transitive, but not symmetric.
(b) Let S be the set of all lines in the plane and let R be the relation “is parallel to.”
It is reflexive (if we agree that a line is parallel to itself), symmetric, and transitive.
(c) Let S be the set of all people who live in Chicago, and suppose that two people x and y
are related by R if x lives within a mile of y.
It is reflexive and symmetric, but not transitive.

10. that are related to a particular element.
Given an equivalence relation R on a set S, it is natural to group together all the elements
that are related to a particular element.
More precisely, we define the equivalence class (with respect to R  ) of x  S to be the set
Ex = { y  S : y R x}.
Since R is reflexive, each element of S is in some equivalence class.
That is, if two
equivalence classes overlap, they must be equal.
Furthermore, two different equivalence classes must be disjoint.
To see this, suppose that w  Ex  Ey.  x y w Ey Ex  x
We claim that Ex = Ey.
For any x  Ex we have x R x.
But w  Ex, so w R x
and, by symmetry, x R w.
Also, w  Ey, so w R y.
Using transitivity twice, we have x R y.
This implies x  Ey and Ex  Ey.
The reverse inclusion follows in a similar manner.
Thus we see that an equivalence relation R on a set S breaks S into disjoint pieces in a natural way.
These pieces are an example of a partition of the set.

11. Definition
A partition of a set S is a collection P of nonempty subsets of S such that
(a) Each x  S belongs to some subset A  P .
(b) For all A, B  P , if A  B, then A  B = .
A member of P is called a piece of the partition.
Example
Let S = {1, 2, 3}. Then the collection P = {{1}, {2}, {3}} is a partition of S.
The the collection P = {{1, 2}, {3}} is also a partition of S. 1 2 3 S 1 2 3 S
P = {{1}, {2}, {3}}
P = {{1, 2}, {3}}
The collection P = {{1, 2}, {2, 3}} is not a partition of S because {1, 2} and {2, 3}
are not disjoint.

12. Not only does an equivalence relation on a set S determine a partition of S,
but the partition can be used to determine the relation.
Theorem
Let R be an equivalence relation on a set S. Then { Ex : x  S} is a partition of S.
The relation “belongs to the same piece as” is the same as R.
Conversely, if P is a partition of S, let P be defined by x P y iff x and y are in the same
piece of the partition. Then P is an equivalence relation and the corresponding partition
into equivalence classes is the same as P .
Proof: Let R be an equivalence relation on S. We have already shown that {Ex : x  S}
is a partition of S.
Now suppose that P is the relation “belongs to the same piece
(equivalence class) as.”
Then
x P y iff x, y  Ez for some z  S.
iff x R z and y R z for some z  S.
iff x R z and z R y for some z  S.
iff x R y
Thus, P and R are the same relation.

13. Not only does an equivalence relation on a set S determine a partition of S,
but the partition can be used to determine the relation.
Theorem
Let R be an equivalence relation on a set S. Then {Ex : x  S} is a partition of S.
The relation “belongs to the same piece as” is the same as R.
Conversely, if P is a partition of S, let P be defined by x P y iff x and y are in the same
piece of the partition. Then P is an equivalence relation and the corresponding partition
into equivalence classes is the same as P .
Proof:
Conversely, suppose that P is a partition of S and let P be defined by x P y iff x and y
are in the same piece of the partition.
Clearly, P is reflexive and symmetric.
To see that P is transitive, suppose that x P y and y P z.
Then y  Ex  Ez.
But this implies that Ex = Ez by the contrapositive of (b), so x P z.
Finally, the equivalence classes of P correspond to the pieces of P because of the way
P was defined. 