Copyright © Cengage Learning. All rights reserved. 6 SETS AND COUNTING Copyright © Cengage Learning. All rights reserved.

6.1 Sets and Set Operations Copyright © Cengage Learning. All rights reserved.

Set Operations

Set Operations The shaded portion of the Venn diagram (Figure 2) depicts the set A  B. Set union A  B Figure 2

Example 7 If A = {a, b, c} and B = {a, c, d}, then A  B = {a, b, c, d}.

Set Operations The shaded portion of the Venn diagram (Figure 3) depicts the set A  B. Set intersection A  B Figure 3

Example 8 Let A = {a, b, c}, and let B = {a, c, d}. Then A  B = {a, c}.

Example 9 Let A = {1, 3, 5, 7, 9}, and let B = {2, 4, 6, 8, 10}. Then A  B = Æ.

Set Operations The two sets of Example 9 have empty, or null, intersection. In general, the sets A and B are said to be disjoint if they have no elements in common—that is, if A  B = Æ.

Example 10 Let U be the set of all students in the classroom. If M = {x  U | x is male} and F = {x  U | x is female}, then F  M = Æ, so F and M are disjoint.

Set Operations The shaded portion of the Venn diagram (Figure 4) shows the set Ac. Set complementation Figure 4

Example 11 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and let A = {2, 4, 6, 8, 10}. Then Ac = {1, 3, 5, 7, 9}.

Set Operations The following rules hold for the operation of complementation.

Set Operations The operations on sets satisfy the following properties.

Set Operations Two additional properties, referred to as De Morgan’s Laws, hold for the operations on sets. Equation (1) states that the complement of the union of two sets is equal to the intersection of their complements. Equation (2) states that the complement of the intersection of two sets is equal to the union of their complements.

Example 12 Using Venn diagrams, show that (A  B) c = Ac  B c Solution: (A  B) c is the set of elements in U but not in A  B and is therefore the shaded region shown in Figure 5. (A  B) c Figure 5

Example 12 – Solution Next, Ac and B c are shown in Figure 6a–b. cont’d Next, Ac and B c are shown in Figure 6a–b. Their union, Ac  B c, is easily seen to be equal to (A  B) c by referring once again to Figure 5. (a) (b) Ac  B c is the set obtained by joining (a) and (b). Figure 6

Example 13 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 4, 8, 9}, and B = {3, 4, 5, 6, 8}. Verify by direct computation that (A  B) c = Ac  Bc.

Example 13 – Solution A  B = {1, 2, 3, 4, 5, 6, 8, 9}, so (A  B)c = {7, 10}. Moreover, Ac = {3, 5, 6, 7, 10} and Bc = {1, 2, 7, 9, 10}, so Ac  Bc = {7, 10}. The required result follows.

Practice p. 330 Self-Check Exercises #2