6. Since there are two statement variables, the truth table will have 4 rows. Start by filling in the four different combinations of truth values for p and q.
| p | q | p L q | ~p | ~q | p L ~q | ~p V (p L ~q) | (p L q) V (~p V (p L ~q)) | ||
| T | T | ||||||||
| T | F | ||||||||
| F | T | ||||||||
| F | F |
Now fill in columns 3, 4 and 5 using the truth tables for L and ~.
| p | q | p L q | ~p | ~q | p L ~q | ~p V (p L ~q) | (p L q) V (~p V (p L ~q)) | ||
| T | T | T | F | F | |||||
| T | F | F | F | T | |||||
| F | T | F | T | F | |||||
| F | F | F | T | T |
| p | q | p L q | ~p | ~q | p L ~q | ~p V (p L ~q) | (p L q) V (~p V (p L ~q)) | ||
| T | T | T | F | F | F | F | |||
| T | F | F | F | T | T | T | |||
| F | T | F | T | F | F | T | |||
| F | F | F | T | T | F | T |
Finally fill in the final column by using columns 3 and 7, and the truth table for V. If the final column contains only T, then you have a tautology; if it contains only F, then you have a contradiction; if it contains a mixture, you have neither.