Doubt in By Case Method to check Tautology

Question

The Truth Value of a compound proposition (if there two pv p,q) is depend on both pv...Then while checking tautology in by case method why we check only one pv’s case like p = true p =false..why we dont check q also?

Comments

If there are two prop vars, p and q. as you mentioned, we check for true case and false case for only one var because it covers every possible case in the truth table, so checking for other var also is redundant. For equival we see if truth values both sides are same, and for tauto, we see if we get lhs T and rhs F in any of the two cases. If there are more vars, we might have to repeat case method.

Answer 1

In by case method, we are trying to check the if the given function/propositional formula has the same behavior/truth values when we fix the truth value of one variable. If they have the same propositional formulas then they will have the same truth values. If we are checking for tautology in 2 variable then the formula would be represented by these 2 variables. So, reducing the formula to just one variable would help in checking in an efficient manner.