Suppose that Pand Q are statements for which P→Q is false. What conclusions (if any) can be made about the truth value of each of the following statements? Briefly explain work. (a) Q (b) P↔Q (recall: P↔Q≡ (P→Q)∧(Q→P)) (c) ¬P→Q

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 10TFE
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Suppose that Pand Q are statements for which P→Q is false. What conclusions (if any) can be made about the truth value of each of the following statements? Briefly explain work.

(a) Q

(b) P↔Q (recall: P↔Q≡ (P→Q)∧(Q→P))

(c) ¬P→Q

Expert Solution
Step 1

Truth table for PQ is:

P Q PQ
T T T
T F F
F T T
F F T

In case PQ is false, which means P is True and Q is False.

If P is true and Q is false, then PQ is False.

¬P is False if P is True.

Now , ¬P is False and Q is also False , so ¬PQ is True.

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