GATE CS 2009 Q23 — Propositional and First Order Logic

The statement "Gold and silver ornaments are precious" means that both of the following are true:

1. All gold ornaments are precious: $\forall x (G(x) \implies P(x))$
2. All silver ornaments are precious: $\forall x (S(x) \implies P(x))$

Combining these two conditions with a logical AND ($\wedge$):

Since the universal quantifier $\forall x$ distributes over conjunction, we can write this as:

Using the logical equivalence $(A \implies C) \wedge (B \implies C) \equiv (A \vee B) \implies C$:

Why other options are incorrect:

• Option A: $\forall x(P(x) \implies (G(x) \wedge S(x)))$ states that "If an object is precious, then it must be both gold and silver simultaneously," which is incorrect.
• Option B: $\forall x((G(x) \wedge S(x)) \implies P(x))$ states that "If an object is both gold and silver, then it is precious," which does not guarantee that an ornament made of only gold (or only silver) is precious.
• Option C: Uses the existential quantifier $\exists x$, which only asserts the existence of at least one such ornament, whereas the original statement is a universal generalization.