# Proving an implication

1. Mar 20, 2013

### John112

A: (p => ~q) ^ (p v q)
B: ~p v q

does A => B (A implies B) ?
does B => A ( B implies A) ?

I did the truth tables for each:

A => B:
http://www4c.wolframalpha.com/input/?i=((p+=>+NOT+q)+AND+(p+OR+q))+=>+(NOT+p+OR+q)

B => A:
http://www.wolframalpha.com/input/?i=(NOT+p+OR+q)+=>+((p+=>+NOT+q)+AND+(p+OR+q))+.

for A to logically imply B or for B to logically imply A, does it need to be a tautology? Meaning if A=> B or B=> A , on the truth tables should it be All Ts? or should the final column of the truh table of A=> B or B => A resemble the truth table of an implication?

Last edited: Mar 20, 2013
2. Mar 20, 2013

### Staff: Mentor

For the implication A => B, the only False you can have is when A is True and B is False. For all other combinations of truth values for A and B, the implication is considered to be True.
For the implication B => A, the only False you can have is when B is True and A is False. For all other combinations of truth values for B and A, the implication is considered to be True.

3. Mar 20, 2013

### John112

So does A logically imply B? or does B logically implies A?

4. Mar 20, 2013

### Staff: Mentor

Make a truth table with five columns, one each for p, q, (p => ~q) ^ (p v q), ~p v q, and ((p => ~q) ^ (p v q)) => ~p v q. You can say that A => B if the only false value you get in the fifth column is in the row where there's a T in the third column and an F in the fourth column.

Similar idea for B => A.