4.1 APPLICATIONS OF LOGIC TO SET THEORY--SOME PROOFS 123- Proceed as in Examples 6 and 7 to show that, for any three sets X, Y, and Z
*(a) If X E Y and X E 2, then X c Y n Z
(6) If X E Z and Y E Z, then X u Y c Z. [Note: For (b), see the remarks
following Example 8 about division of an argument into cases.]
- Proceed as in Example 9 to show that, for any two sets A and B, A E B if and
only if A u B = B. (Again, keep in mind the possibility of division of an argument
into cases.) - Proceed as in Example 11 to show that, for any set X, X n 0 = 0.
- Follow the approach of Example 10 to prove that, for any sets A, B, and C:
(a) A-B=AnB' (b) AnB=BnA
(c) AuB=BuA (d) A n (B n C) = (A n B) n C
(e) A u (B v C) = (A u B) u C (f) A u (B n C) = (A u B) n (A u C)
(g) (A n B)' = A' u B' (h) (A u B)' = A' n B' - Follow the approach of Examples 1 through 3 (relying directly on logical
principles) to prove that
(a) For any set A:
(i) A r U
(iii) A = A"
(v) AnA=A
(ii) A = A
(iv) A u A =A(b) (i) (25' = U [Hint: Use (i) of (a) for one part of this argument.]
(ii) U' = 0- (a) Prove that if A and B are any sets such that A E B, then B' s A'. (Hint:
Start by letting x E B' and suppose the opposite of what you need to conclude.
Reread the solution to Example 12.)
*(b) Use (a), together with the fact that A = A" for any set A, to prove the con-
verse of (a); that is, given sets A and B, if B' E A', then A E B.
(c) Use the results in (a) and (b), together with the result of Example 3, to show
that if A and B are any two sets, then A = B if and only if A' = B'. - (a) Prove that, for any sets A and B, if A E B, then A' u B = U. [Hint: One
approach is to use the result of Example 12, one of De Morgan's laws, and the
fact that 0' = U.]
(b) Prove that, for any set A, A u A' = U. [Hint: Use (a).] - Prove that, for any sets A and B:
*(a) AnB=UifandonlyifA=UandB=U
(b) A u B # (25 if and only if either A # (25 or B # 0
(c) If A = U or B = U, then A u B = U
(d) If A n B # a, then A # (25 and B # (25 (Note: Check the connection
between the statements in Exercise 9 and various parts of Theorems 1 and 2,
Article 3.3.)
- As suggested in Article 1.2, a set theoretic definition of the "ordered pair" concept
exists. We may define (a, b) to be the set {{a), {a, b)), where a and b are any objects.