Operations on Sets 17
Definition 2. Let A and B be sets. The intersection of A and B, denoted by A n B, is
{x :xE A and x E B)
The intersection of A and B is shaded in Figure 1.6.
U
Figure 1.6 A n B.
Example 2.
(a) {1,2, 3} n {3,4,51 = 13).
(b) {1, 2, 31 n 14, 5, 6) = 0.
(c) N•nZ =N.
(d) For any set A, A n 0 =0.
(e) {1, 2, 3) n {{1, 2, 3}1 = 0. (The first set has three elements, 1, 2, and 3, whereas the
second set has only one element, (1, 2, 31.)
Theorem 2 proves some fundamental results about set intersection. Like set union, set
intersection satisfies the commutative and associative laws.
Theorem 2. Let A, B, and C be sets.
(a) ANA=A.
(b) ANBCAandANBCB.
(c) A n B = B n A. (Commutative Law for Intersection)
(d) A n (B n C) = (A n B) n C. (Associative Law for Intersection)
(What the proof entails.) Parts (a) and (b) follow directly from the definition of inter-
section. Part (c) says that the order in which the intersection of two sets is formed does not
matter. Part (d) states that A n B n C makes sense even without parentheses.
Proof (c) Again, follow Template 1.5 (Set Equality). Prove that (i) A n B C B n A
and (ii) B n A C A n B. For (i), follow the template for proving one set is a subset of
another. That is, assume x E A n B, and show x E B n A.
Suppose x E A n B. Then, x E A and x E B. Equivalently, x E B and x E A, since
no order is implied by the word and. Therefore, x E B n A. The proof of (ii) is analogous.
(d) This part is left as an exercise for the reader. 0
The distributive laws for addition and multiplication for real numbers have analogues
with the operations of union and intersection with sets, as Theorem^3 shows.
Theorem 3. (Set Distributivity) Let A, B, and C be sets. Then:
(a) A U (B n C) = (A U B) n (A U C). (Distributive Law for Union)
(b) A A (B U C) = (A n B) U (A n C). (Distributive Law for Intersection)
