Final answer:
To show that ((p ∨ q) ∨ ((∼ p) ∧ (∼ q))) is a tautology for any propositions p and q, we can use truth tables.
Step-by-step explanation:
To show that ((p ∨ q) ∨ ((∼ p) ∧ (∼ q))) is a tautology for any propositions p and q, we can use truth tables.
A tautology is a statement that is always true, regardless of the truth values of its variables.
In this case, we have two variables: p and q. We need to consider all possible combinations of truth values for these variables and evaluate the statement for each combination.
Since there are 2 variables, we have 2^2 = 4 possible combinations of truth values.
Using a truth table, we can calculate the truth values of ((p ∨ q) ∨ ((∼ p) ∧ (∼ q))) for all possible combinations of truth values for p and q.
After evaluating each combination, we will find that the statement is always true, regardless of the truth values of p and q.