Verify whether (~p^(pvq))→ q is a tautology using logical transformations (not using truth tables)

To verify whether the statement (~p^(pvq))→ q is a tautology, we will use logical transformations.

Step 1: We will start by simplifying the expression inside the parentheses:

~p^(pvq)

Step 2: Distribute the conjunction (^) over disjunction (v):

(~p^pv) v (~p^q)

Step 3: Simplify the first conjunction (~p^pv):

(~p^pv) ≡ (~p) v p

Step 4: Negate the first term (~p):

(~p) ≡ p -> false, so ~p is equivalent to false.

Step 5: Simplify the second conjunction (~p^q):

(~p^q) ≡ false v q ≡ q

Step 6: Substitute the simplified expressions back into the original expression:

(~p^pv) v (~p^q) ≡ false v q

Step 7: Simplify the disjunction (false v q):

false v q ≡ q

Step 8: Substitute the simplified expression back into the original statement:

(~p^(pvq))→ q ≡ q -> q ≡ q

Answer: The statement (~p^(pvq))→ q simplifies to q, indicating that it is a tautology.