Chapter 2 Sets and Functions

Section 2.2 Relations

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.

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 …

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. 

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 [ )

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 1 < 3 is true, 2 < 7 is true, 5 < 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.

Example 2.2.8 Returning to Example 2.2.5 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].

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 2.2.10 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.

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.

Definition 2.2.12 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 2.2.13 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.

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 2.2.17 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.

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 2.2.17 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 2.2.12(b), so x P z. Finally, the equivalence classes of P correspond to the pieces of P because of the way P was defined. 