Construct a truth table for q ^ ~p
I'll walk yoiu through it. Pay attention:
You begin with this:
p q ~p q ^ ~p
Case 1
Case 2
Case 3
Case 4
There are four cases
Case 1: p has truth value T, q has truth value T
Case 2: p " " " T, q " " " F
Case 3: p " " " F, q " " " T
Case 4: p " " " F, q " " " F
Every time I complete a case I will color it red:
For case 1, temporarily replace the 3 p's with T's and the 2 q's with T's
p q ~p q ^ ~p
Case 1 T T ~T T ^ ~T
Case 2
Case 3
Case 4
Then erase the 2 ~T's and replace them by F's
p q ~p q ^ ~p
Case 1 T T F T ^ F
Case 2
Case 3
Case 4
Now use the rule for ^ which says we replace T ^ T by T and anything else by F
So we erase the T ^ F and replace it by F
p q ~p q ^ ~p
Case 1 T T F F
Case 2
Case 3
Case 4
and we are done with Case 1, so we color it red.
p q ~p q ^ ~p
Case 1 T T F F
---------------------------------
For case 2, temporarily replace the 3 p's with T's and the 2 q's with F's
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F ~T F ^ ~T
Case 3
Case 4
Then erase the 2 ~T's and replace them by F's
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F ^ F
Case 3
Case 4
Now use the rule for ^ which says we replace T ^ T by T and anything else by F
So we erase the F ^ F and replace it by F
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F
Case 3
Case 4
and we are done with Case 2, so we color it red.
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F
Case 3
Case 4
---------------------------------
For case 3, temporarily replace the 3 p's with F's and the 2 q's with T's
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F
Case 3 F T ~F T ^ ~F
Case 4
Then erase the 2 ~F's and replace them by T's
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F
Case 3 F T T T ^ T
Case 4
Now use the rule for ^ which says we replace T ^ T by T and anything else by F
So we erase the T ^ T and replace it by T
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F
Case 3 F T T T
Case 4
and we are done with Case 3, so we color it red.
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F
Case 3 F T T T
Case 4
---------------------------------
For case 4, temporarily replace the 3 p's with F's and the 2 q's with T's
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F
Case 3 F T T T
Case 4 F F ~F F ^ ~F
Then erase the 2 ~F's and replace them by T's
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F
Case 3 F T T T
Case 4 F F T F ^ T
Now use the rule for ^ which says we replace T ^ T by T and anything else by F
So we erase the F ^ T and replace it by F
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F F F
Case 3 F T T T
Case 4 F F T F
and we are done with Case 4, so we color it red.
p q ~p q ^ ~p
Case 1 T T F F
Case 2 T F ~T F
Case 3 F T T T
Case 4 F F T F
And we're done with the whole truth tablw. You can learn to do
these case by case without erasing anything, by doing those steps
in your head. It isn't hard.
Edwin