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GATE 2014 CS (CS1) – Question 11

Engineering Mathematics · Discrete Mathematics: Propositional and First Order Logic · 1 mark · Multiple choice

“Not all that glitters is gold.” Predicate glitters(x) means x glitters, and gold(x) means x is gold. Which formula represents this statement?

  1. $\forall x:\operatorname{glitters}(x)\Rightarrow\neg\operatorname{gold}(x)$
  2. $\forall x:\operatorname{gold}(x)\Rightarrow\operatorname{glitters}(x)$
  3. $\exists x:\operatorname{gold}(x)\land\neg\operatorname{glitters}(x)$
  4. $\exists x:\operatorname{glitters}(x)\land\neg\operatorname{gold}(x)$

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Correct answer: (D) $\exists x:\operatorname{glitters}(x)\land\neg\operatorname{gold}(x)$

Explanation

Negate the universal claim ∀x(glitters(x)⇒gold(x)). This gives ∃x(glitters(x)∧¬gold(x)), meaning at least one glittering object is not gold. Hence **D**. It does not claim that every glittering object fails to be gold.