Page 57 - HOW TO PROVE IT: A Structured Approach, Second Edition

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The Conditional and Biconditional Connectives 43
(a) What’s wrong with the following diagram? (Hint: Where’s the set
(A ∩ D) \ (B ∪ C)?)

(b) Can you make a Venn diagram for four sets using shapes other than
circles?
11. (a) Make Venn diagrams for the sets (A ∪ B) \ C and A ∪ (B \ C). What
can you conclude about whether one of these sets is necessarily a
subset of the other?
(b) Give an example of sets A, B, and C for which (A ∪ B) \ C = A ∪
(B \ C).
∗ 12. Use Venn diagrams to show that the associative law holds for symmetric
difference; that is, for any sets A, B, and C, A (B C) = (A B) C.
13. Use any method you wish to verify the following identities:
(a) (A B) ∪ C = (A ∪ C) (B \ C).
(b) (A B) ∩ C = (A ∩ C) (B ∩ C).
(c) (A B) \ C = (A \ C) (B \ C).
14. Use any method you wish to verify the following identities:
(a) (A ∪ B) C = (A C) (B \ A).
(b) (A ∩ B) C = (A C) (A \ B).
(c) (A \ B) C = (A C) (A ∩ B).
15. Fill in the blanks to make true identities:
(a) (A B) ∩ C = (C \ A) .
(b) C \ (A B) = (A ∩ C) .
(c) (B \ A) C = (A C) .

1.5. The Conditional and Biconditional Connectives

It is time now to return to a question we left unanswered in Section 1.1. We
have seen how the reasoning in the first and third arguments in Example 1.1.1
can be understood by analyzing the connectives ∨ and ¬. But what about the