Answer to Question #245877 in Discrete Mathematics for dev

Question

1. Let p and q be the propositions

p: It snowed.

q: Eve goes skiing.

Express each of these propositions using p and q and logical connectives.

(a) It snowed but Eve does not go skiing.

(b) Whenever Eve goes skiing, it snowed.

(c) It having snowed is sufficient for Eve to go skiing.

(d) For Eve to go skiing, it is necessary that it snowed.

(e) If it didn’t snow, then Eve does not go skiing.

(f) Whenever Eve doesn’t go skiing, it has not snowed.

2. For the propositions in Problem 1, identify all pairs of propositions that are logically

equivalent. Justify your answer.

Answer 1

1.

a) $p\land \neg q$

b) $q\to p$

c) $p\to q$

d) $q\to p$

e) $\neg p\to \neg q$

f) $\neg q\to \neg p$

2.

Using logical equivalences:

$p\to q\equiv \neg q\to \neg p$

$q\to p\equiv \neg p\to \neg q$

So, b), d), e) are logically equivalent; and c), f) are logically equivalent.