GATE CS 2000 Q2.7 — Propositional and First Order Logic

We are given two equivalences:

1. $a \iff (b \lor \neg b)$
   Since $b \lor \neg b$ is a tautology (always True), the proposition $a$ must always be True ($a \equiv \text{True}$).
2. $b \iff c$
   This means $b$ and $c$ always have the same truth value ($b \equiv c$).

Now, let us evaluate the given formula:

Substituting $a \equiv \text{True}$ into the formula:

Since $b \equiv c$, we can substitute $c$ with $b$:

Using the implication identity $P \to Q \equiv \neg P \lor Q$:

Since $\neg b \lor b \equiv \text{True}$:

Thus, the truth-value of the formula is always True.