You can
put this solution on YOUR website!# 1
Is there supposed to be a 'p' somewhere in there? Please double check it.
# 2
I'll do the second one to get you going. If this isn't enough, either ask me or repost.
Start with a blank table with 4 rows and with the headers of p, q, ~p, ~p ^ q, (~p ^ q) -> p. The headers are simply smaller pieces of (~p ^ q) -> p
Fill in T, T, F, F in the first column and T, F, T, F in the second. This will exhaust all of the possible truth combinations of p and q
| p | q | ~p | ~p ^ q | (~p ^ q) -> p | | T | T | | | |
| T | F | | | |
| F | T | | | |
| F | F | | | |
Negate the first column p to get ~p for the third column
| p | q | ~p | ~p ^ q | (~p ^ q) -> p | | T | T | F | | |
| T | F | F | | |
| F | T | T | | |
| F | F | T | | |
Recall that p ^ q is only true when BOTH p and q are true. Otherwise it is false. So ~p ^ q is only true when the corresponding entries of the columns ~p and q are both T, or otherwise it's false. Use this info to fill in the fourth column.
| p | q | ~p | ~p ^ q | (~p ^ q) -> p | | T | T | F | F | |
| T | F | F | F | |
| F | T | T | T | |
| F | F | T | F | |
Now remember that p -> q is only false when p is true, but q is false (otherwise, it is true). This is like me claiming "if it rains, then it gets wet". If it does rain, but it does not get wet, then my statement is false. Fortunately for us, much of the fourth column is false which will make much of the last column true. Let's now use this information to complete the table.
| p | q | ~p | ~p ^ q | (~p ^ q) -> p | | T | T | F | F | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | F | T |
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