Question 519282
<pre>
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
<font color="red">Case 1   T   T    F     F</font>

---------------------------------

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
<font color="red">Case 1   T   T    F     F</font>
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
<font color="red">Case 1   T   T    F     F</font>
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
<font color="red">Case 1   T   T    F     F</font>
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
<font color="red">Case 1   T   T    F     F
Case 2   T   F    F     F</font>
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
<font color="red">Case 1   T   T    F     F
Case 2   T   F    F     F</font>
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
<font color="red">Case 1   T   T    F     F
Case 2   T   F    F     F</font>
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
<font color="red">Case 1   T   T    F     F
Case 2   T   F    F     F</font>
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
<font color="red">Case 1   T   T    F     F
Case 2   T   F    F     F
Case 3   F   T    T     T</font>
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
<font color="red">Case 1   T   T    F     F
Case 2   T   F    F     F
Case 3   F   T    T     T</font>
Case 4   F   F   ~F   F ^ ~F 

Then erase the 2 ~F's and replace them by T's

         p   q   ~p   q ^ ~p
<font color="red">Case 1   T   T    F     F
Case 2   T   F    F     F
Case 3   F   T    T     T</font>
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
<font color="red">Case 1   T   T    F     F
Case 2   T   F    F     F
Case 3   F   T    T     T</font>
Case 4   F   F    T     F


and we are done with Case 4, so we color it red.

         p   q   ~p   q ^ ~p
<font color="red">Case 1   T   T    F     F
Case 2   T   F   ~T     F
Case 3   F   T    T     T
Case 4   F   F    T     F</font>


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</pre>