Discrete Mathematics

Question

Determine the truth values of each proposition below.

(a) 1 + 1 = 3 if and only if 2 + 2 = 3.

(b) If 1 + 1 = 2 or 1 + 1 = 3, then 2 + 2 = 3 and 2 + 2 = 4.

(c) If squirrels play badminton, then cats can't fly.

Answer 1

All these question involve implication and it helps to realize that

p⇒q is just shorthand for ¬p∨q

(a)

"p if and only if q" is short for

p⇒q and q⇒p, which is short for

¬p∨q and ¬q∨p

In the problem (a) both p and q are false, therefore

both implications are true, therefore the whole equivalence is true.

(b)

Here we have p⇒q, where

p is an OR of two statements, one of which is true, therefore p is true, and

q is the AND of two statements, one of which is false, therefore q is false.

That makes the whole implication false.

(c)

Again we have p⇒q,

but I am not going to pass judgement on biological capabilities of either squirrels or cats.

Assuming that p is false and q is true, the whole implication is true.