Final answer:
The given formulas pv~p, p->p, and p<->p are all tautologies, meaning they are always true regardless of the values of the variables involved.
Step-by-step explanation:
The given formulas pv~p, p->p, and p<->p are all tautologies, meaning they are always true regardless of the values of the variables involved. A tautology is a formula that is true in every interpretation (or truth assignment) to its proposition variables. To show this, we can analyze the truth tables for each formula:
For pv~p, the truth table represents two cases where p is true (T) and one where p is false (F). Since the formula is of the form 'statement or not p', it will be true for every case.
The truth table for p->p consists of two cases where p is true (T), and one case where p is false (F). The formula represents 'p implies p', and since an implication is always true when the antecedent is false (F->T), it will be true for every case.
p<->p also has two cases where p is true (T) and one case where p is false (F). The formula represents 'p if and only if p', and since it is an equivalence relationship, it will be true for every case.