### Question Description

Image text transcribed for accessibility: Which of the following statements are true for all propositions p and q? (True/False)If p q, then p is true or q is false. (True/False)If p q, then p is false or q is true. (True/False)If p q. then (q p). (True/False)If p q then p q. or q p. (True/False) Either p q. or (p q).

## Explanation & Answer

p->q is the same as "q or not-p"

1.) False.

If for example p is false and q is true, then:

p->q is true,

BUT

neither "p is true" nor "q is false" is true.

2.) True.

By definition of p->q above.

3.) False.

If for examples p and q are both true, then:

p->q is true,

BUT

~(q->p) is false.

[Moral: Just because p->q does not mean that q cannot imply p. An easier example is when p and q are the same statement!]

4.) True.

If BOTH p and q are true, then both implications hold. If EITHER is false, say p is false, then p->q is automatically true. (Similarly, if q is false, then q->p is automatically true.)

5.) True.

This is a tautology. Let r = (p->q), then the statement becomes: r or ~r, which is always true.