Verify by using the laws of logic whether the following statement is a Tautology or not | Query Point Official

Verify Whether a Logical Statement Is a Tautology Using Laws of Logic

Problem Statement

Verify using laws of logic whether the following statement is a tautology (always true for all truth values):

`(~(~q → p)) → ~q`

Step 1: Rewrite the Inner Implication

Using the equivalence A → B ≡ ~A ∨ B:

~q → p ≡ q ∨ p

Negate the implication:

~(~q → p) ≡ ~(q ∨ p)

Step 2: Apply De Morgan's Law

~(q ∨ p) ≡ ~q ∧ ~p

Thus, ~(~q → p) ≡ ~q ∧ ~p

Step 3: Substitute Back Into the Main Statement

(~(~q → p)) → ~q ≡ (~q ∧ ~p) → ~q

Step 4: Convert Implication to OR

Using A → B ≡ ~A ∨ B:

(~q ∧ ~p) → ~q ≡ ~(~q ∧ ~p) ∨ ~q

Step 5: Apply De Morgan's Law Again

~(~q ∧ ~p) ≡ q ∨ p

So the statement becomes:

(q ∨ p) ∨ ~q

Step 6: Simplify Using Associative Law

(q ∨ p) ∨ ~q ≡ (q ∨ ~q) ∨ p

Step 7: Apply the Law of the Excluded Middle

q ∨ ~q ≡ True

(q ∨ ~q) ∨ p ≡ True ∨ p ≡ True

Conclusion

Since the statement always evaluates to True, the logical statement (~(~q → p)) → ~q is a tautology.

FAQ

What does “tautology” mean?

A tautology is a logical expression that evaluates to true for all values of its variables. For example, `P ∨ ~P` is always true.

Why use laws of logic instead of a truth table?

Laws of logic allow symbolic simplification and proof of equivalence without listing every possible truth assignment. Creativity and efficiency can make solving complex expressions easier.

Where else are such problems useful?

These techniques are widely used in discrete mathematics, digital logic design, programming semantics, and proofs in computer science and mathematics.

Related Topics

See Mathematics Notes & MCQs for more problems on logic, truth tables, and equivalence laws.