Animated Solution for Mathematics - Sets and Relations: Which of the following statement is a tautology ?

Welcome, fellow explorer of the mathematical universe! Today, we are embarking on a journey into the heart of logical reasoning to identify a tautology—a statement that remains true under every possible circumstance.

We have two primary paths: the brute force of truth tables or the elegance of logical identities. We will utilize the latter, focusing on three key concepts:

De Morgan's Law: Allows us to flip operators when negating. Implication Law: Transforms the expression $X\to Y$ into $\sim X\vee Y$. Identity Laws:* Simplify expressions to a constant $T$.

Let us focus our attention on the first candidate: $\sim (p\vee \sim q)\to (p\vee q)$.

First, we tackle the antecedent: $\sim (p\vee \sim q)$. Applying De Morgan's Law, we distribute the negation to obtain:

Next, we address the implication arrow. Using the Implication Law, $X\to Y\equiv \sim X\vee Y$, our expression becomes:

Now, we simplify the first part: $\sim (\sim p\wedge q)$. Applying De Morgan's Law again, we get:

We are now left with the full expression:

Because all our operators are OR, the Associative Law allows us to drop the brackets entirely:

Using the Commutative Law, we rearrange the terms to group them:

Look at what we have created! The Idempotent Law tells us that $p\vee p\equiv p$, and the Complement Law tells us that $\sim q\vee q\equiv T$.

Our expression has shrunk to:

Finally, the Domination Law dictates that anything OR True is always True. Thus:

We have arrived at the constant True. We have proven that Option 1 is indeed a tautology. It is not just a statement; it is a logical necessity.