Figure 7

these pairs as corresponding points in labeled rows and columns, in the manner shown on the right in the figure. The reader should list or draw T × S and note that T × S  S × T.

(b) If S = {1, 2, 3, 4}, then S² == S  S has sixteen ordered pairs; see Figure 8. Note that (2, 4)  (4, 2); these ordered pairs involve the same two numbers, but in different orders. In contrast, the sets {2, 4} and {4, 2} are the same. Also note that (2, 2) is a perfectly good ordered pair in which the first element happens to equal the second element. On the other hand, the set {2, 2} is just the set {2} in which 2 happens to be written twice.

(1, 4) (2, 4) (3, 4) (4, 4) 4 ○ ○ ○ ○ (1, 3) (2, 3) (3, 3) (4, 3) 3 ○ ○ ○ ○ (1, 2) (2, 2) (3, 2) (4, 2) 2 ○ ○ ○ ○ (1, 1) (2, 1) (3, 1) (4, 1) 1 ○ ○ ○ ○ 1 2 3 4 List of {1, 2, 3, 4}2 Picture of {1, 2, 3, 4}2

Figure 7

Both uses of this notation are standard, however. Fortunately, the intended meaning is almost always clear from the context. ■

For any finite set S, we write |S| for the number of elements in the set. Thus |S| = |T| precisely when the finite sets S and T are of the same size. Observe that

|| = 0 and |{l, 2, ..., n)}| = n for n  P.

Moreover, |S  T| = |S||T|. You can see where the notation  for the product of two sets came from. It turns out that |P(S)| = 2^|S|, so some people also use the notation 2^S for P(S).

We can define the product of any finite collection S₁, S₂, ...,S_n, of sets. The product set S₁  S₂   S_n consists of all ordered n-tuples (s₁, s₂, ..., s_n) where s₁  S₁, s₂  S₂, etc. That is,

S₁  S₂   S_n = {(s₁, s₂, ..., s_n) : s_k  S_k for k = 1, 2,..., n}.

Just as with ordered pairs, two ordered n-tuples (s₁, s₂, ..., s_n) and (t₁, t₂, ..., t_n) are regarded as equal if all the corresponding entries are equal: s_k = t_k for k = 1, 2,..., n. If the sets S₁, S₂, ...,S_n are all equal, to S say, we may write Sⁿ for the product S₁  S₂   S_n.

Exercises 1.2

1. Let U = {1, 2, 3, 4, 5, ..., 12}, A = {1, 3, 5, 7, 9, 11}, B = {2, 3, 5, 7, 11}, C = {2, 3, 6, 12} and D = {2, 4, 8}. Determine the sets

(a) A  B (b) A  C (c) (A  B)  C^c

(d) A\B (e) C\D (f) B  D

(g) How many subsets of C are there?

2. Let A = {1, 2, 3}, B = {n  P :n is even} and C = {n  P :n is odd}.

(a) Determine A  B, B  C, B  C and B  C.

(b) List all subsets of A.

(c) Which of the following sets are infinite? A  B, A  C, A\C, C\A.

3. In this exercise the universe is R. Determine the following sets.

(a) [0,3]  [2,6] (b) [0, 3]  [2,6] (c) [O, 3]\[2,6]

(d) [0,3]  [2,6] (e) [0. 3]^c (f) [0,3]  

4. Let  = {a, b}, A = {a, b, aa, bb, aaa, bbb}, B = {w  ^*: length(w)  2} and C= {w  ^:length(w)  2}.

(a) Determine A  C, A\C, C\A and A  C.

(b) Determine A  B, B  C, B  C and B\A.

(c) Determine ^*\B, \B and \C.

(d) List all subsets of .

(e) How many sets are there in P()?

5. In this exercise the universe is ^* where  = {a, b}. Let A, B and C be as in Exercise 4. Determine the following sets.

(a) B^c  C^c (b) (B  C)^c (c) (B  C)^c

(d) B^c  C^c (e) A^c  C (f) A^c  B^c

(g) Which of these sets are equal? Why?

6. The following statements involve subsets of some nonempty universal set U. Tell whether each is true or false. For each false one, give an example for which the statement is false.

(a) A  (B  C) = (A  B)  C for all A, B, C.

(b) A  B  A  B implies A = B.

(c) (A  )  B = B for all A, B.

(d) A  (  B) = A whenever A  B.

(e) A  B = A^c  B^c for all A, B.

7. For any set A, what is A  A? A  ?

8. Use Venn diagrams to prove the following.

(a) A  (B  C) = (A  B)  (A  C)

(b) A  B  (A  C)  (B  C)

9. Prove the generalized DeMorgan law (A  B  C)^c = A^c  B^c  C^c. Hint:

First apply the DeMorgan law 9b to the sets A and B  C. The elementwise method can be avoided.

10. Prove the following without using Venn diagrams.

(a) A  B  A and A  A  B for all sets A and B.

(b) If A  B and A  C, then A  B  C.

(c) If A  C and B  C, then A  B  C.

(d) A  B if and only if B^c  A^c.

11. Let A = {a, b, c} and B = {a, b, d}.

(a) List or draw the ordered pairs in A  A.

(b) List or draw the ordered pairs in A  B.

(c) List or draw the set {(x, y)  A  B :x = y}.

12. Let S = {0, 1, 2, 3, 4} and T = {0, 2, 4}.

(a) How many ordered pairs are in S  T? T  S?

(b) List or draw the elements in {(m, n)  S  T :m < n}.

(c) List or draw the elements in {(m, n)  T  S :m < n}.

(d) List or draw the elements in {(m, n)  S  T :m + n  3}.

(e) List or draw the elements in {(m, n)  T  S :mn  4}.

(f) List or draw the elements in {(m, n)  S  S :m + n = 10}.

13. For each of the following sets, list all elements if the set has fewer than seven elements. Otherwise, list exactly seven elements of the set.

(a) {(m, n)  N² :m = n} (b) {(m, n)  N² :m + n is prime} (c) {(m, n)  P² :m = 6} (d) {(m, n)  P² :min{m, n} = 3}

(e) {(m, n)  P² :max{m, n} = 3} (f) {(m, n)  N² : m² = n}

14. Draw a Venn diagram for four sets A, B, C and D. Be sure to have a region for each of the sixteen possible sets such as A  B^c  C^c  D.

Note. In the remaining exercises, you may use any method of proof.

15. Prove or disprove. [A proof needs to be a general argument, but a single counterexample is sufficient for a disproof.]

(a) A  B = A  C implies B = C.

(b) A  B = A  C implies B = C.

(c) A  B = A  C and A  B = A  C imply B = C.

(d) A  B  A  B implies A = B.

(e) A  B = A  C implies B = C.

16. (a) Prove that A  B if and only if A  B = B. This is really two assertions:

“A  B implies A  B = B” and “A  B = B implies A  B.”

(b) Prove that A  B if and only if A  B = A.

17. (a) Show that relative complementation is not commutative; that is, A\B = B\A can

fail.

(b) Show that relative complementation is not associative: (A\B)\C = A\(B\C) can

fail.

(c) Show, however, that (A\B)\C  A\(B\C) for all A, B and C.