Why Read This Book? - Index of

70 Chapter 3 Sets and Their Properties
Proof. Suppose A − B is non-empty. Then there exists x ∈ (A − B). Thus x ∈ A ∩
B C , so that x ∈ A and x ∈ B C . But the existence of such an x is precisely the
negation of Definition 3.1.7, so that A �⊆ B.
Proof by contradiction can come in handy, too.
Theorem 3.2.7 If A is any set, ∅⊆A.
Proof. Suppose there exists a set A such that ∅ �⊆ A. Then there exists x ∈∅such
that x/∈ A. But ∅ contains no elements. This contradicts the definition of ∅. Thus
∅⊆A.
EXERCISE 3.2.8 State the converse of Theorem 3.2.1 and prove it.
EXERCISE 3.2.9 Prove the following.
(a) For all sets A and B,ifA ⊆ B, then A ∪ B = B.
(b) If A ⊆ B and B ⊆ C, then A ⊆ C.
(c) A ∩ AC =∅.
(d) If A ⊆ B, then AC ⊇ BC .
(e) (DeMorgan’s Law) For all sets A and B, (A ∪ B) C = AC ∩ BC .
(f) If A and B are disjoint, then A�B = A ∪ B.
(g) If A ⊆ C and B ⊆ C, then A ∪ B ⊆ C.
(h) If C ⊆ A and C ⊆ B, then C ⊆ A ∩ B.
(i) If A ⊆ B and C ⊆ D, then A ∩ C ⊆ B ∩ D.
(j) If A ⊆ B and C ⊆ D, then A ∪ C ⊆ B ∪ D.
EXERCISE 3.2.10 State and prove the converse of Exercise 3.2.9(f). 3
EXERCISE 3.2.11 Prove that ∩ distributes over ∪ and vice versa.
EXERCISE 3.2.12 Sometimes we can prove certain sets are equal without having
to chase elements back and forth, by appealing to earlier theorems we have
proved. By making appropriate references to certain results from Exercise 3.2.9,
prove the following are true for all sets M, N, S, and T .
(a) If M = N and S = T , then M ∩ S = N ∩ T .
(b) If M = N and S = T , then M ∪ S = N ∪ T .
(c) If M = N, then MC = NC .
3 Try proof by contrapositive or contradiction.