mathematics help

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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).

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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,
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,
~(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.

xunxnna (1456)
UT Austin

Really useful study material!