p^(~p v q)

Put TTFF under the p's and TFTF under the q

p^(~p v q)
T   T   T
T   T   F
F   F   T
F   F   F

Under the ~ before the second p, put the opposite of what 
is under the second p and erase what was under the 2nd p"

p^(~p v q)
T  F    T
T  F    F
F  T    T
F  T    F

Under the v, put F only if there are F's on BOTH sides of it,
otherwise put T.  Then erase what is on both sides of it:

p^(~p v q)
T     T  
T     F  
F     T  
F     T  
 
Under the ^, put T only if there are T's on BOTH sides of it,
otherwise put F.  Then erase what is on both sides of it:

p^(~p v q)
 T      
 F      
 F      
 F

So the truth table ends up TFFF, which means it's true only when
both p and q are true, otherwise it's false.  

Edwin