Verify that the proposition pv~(p^q) is a tantology
To verify if the proposition is a tautology, we need to show that it is true for all possible truth values of the propositional variables p and q.
Let's construct a truth table for the proposition pv~(p^q):
| p | q | p^q | ~(p^q) | pv~(p^q) |
|---|---|-----|--------|----------|
| T | T | T | F | T |
| T | F | F | T | T |
| F | T | F | T | T |
| F | F | F | T | F |
In each row, we evaluate the logical operators in order. The column p^q represents the conjunction (AND) of p and q, and ~(p^q) represents the negation (NOT) of p^q. Finally, pv~(p^q) represents the disjunction (OR) of p and ~(p^q).
From the truth table, we can see that for every combination of truth values for p and q, the proposition pv~(p^q) evaluates to true (T). Therefore, the proposition is a tautology.