5.3 PROOF BY SPECIALIZATION AND DIVISION INTO CASES 175
(c) Prove or disprove the converse of the theorem proved in Example 8, Article
5.2, namely, if A, X, and Y are sets such that A n X c A n Y, then X E Y.
(d) Prove that if A, X, and Y are sets satisfying the properties A n X E A n Y
and A u X c A u Y, then X G Y [recall Exercise 3(a), Article 1-31.
(e) Use the result of (d) to prove that if A, X, and Y are sets satisfying A n X =
A n Y and A u X = A u Y, then X = Y.
(f) Recall from Exercise 10, Article 3.3, the definition of a complement of a set.
Use (e) to show that if a set A has a complement, this complement is unique.
- Prove that if A, B, and X are sets satisfying A u X c B u X and A u X' E
B u X', then A c B.
4. *(a) Prove, by the choose method, that if X, A, and B are sets such that X E B,
then X u (A n B) c (X u A) n B. (Note: The reverse inclusion is true for any
three sets X, A, and B, that is, without the hypothesis X G B. Its proof will be
considered in Article 6.2.)
(b) Show by example that the converse of the theorem in (a) is false. Specifically,
show that there exist sets X, A, and B such that X u (A n B) = (X u A) n B,
but X is not a subset of B.
(c) Prove that if X and B are sets, from a universal set U, satisfying the property
X u (A n B) = (X u A) n B for. every set A from U, then necessarily X G B.
[Note: This result is called a partial converse of the theorem in (a). Hint: For
the proof, use specialization.]
(d) Is there any logical conflict between the results of (b) and (c)? Interpret these
results in the light of Exercise 11, Article 3.4.
(e) Prove that if X and Y are sets having the property that, for every set A (from
the common universal set that also contains X and Y), A n X E A n Y, then
X c Y. [This is a partial converse to the theorem in Example 8, Article 5.2.
Recall Exercise 2(c).]
- (Continuation of Exercise 2, Article 5.2)
(a) Prove that if A and B are sets, then B(A) u P(B) c 9(A u B).
(b) Give an example to show that the reverse inclusion in (a) need not be true.
(c) Prove that, for any sets A and B, if B(A) u B(B) = B(A u B), then either
A E B or B E A. [Hint: Use specialization followed by division into cases. Note
that A u B itself is an element of A u B), since A u B c A u B.]
(d) In view of the results from (a) and (c), what relationship exists between
A) u P(B) and B(A u B) for any two sets A and B, neither of which is a
subset of the other?
6. Prove by the chobse method (in particular, do not use distributivity) that if A
and B are sets, then (A n B) u (A n B') = A.
- Use distributivity and previously proved identities involving union and inter-
section to give a proof by transitivity (as in Article 5.1) that if A, B, and X are
subsets of a universal set U, and:
(a) If X E B, then X u (A n B) = (X u A) n B
*(b) If A n X = B n X and A n X' = B n X', then A = B
(c) IfAuX=BuXandAuX'=BuX',thenA=B
(d) IfAnX=BnXandAuX=BuX,thenA=B
