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Question 261189: Can someone please help me with these truth tables? When I get my tax check I am planning on making a donation to the site because this site has really helped me out alot. This is 3 tables out of a slew I have and I really need help on these, so I can do the rest of my work.
1) (q ^ ~q) <-> q
2)(~p ^ q) -> p
3) q -> (q V ~q)
Thank you soooo much in advance for your time and effort to help me...
Answer by jim_thompson5910(35256)   (Show Source):
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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