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Consider the statement “Not all that glitters is gold” Predicate glitters(x) is true if x glitters and predicate gold(x) is true if x is gold. What which one of the following logical formulate represents the above statement?

Consider the statement “Not all that glitters is gold” Predicate glitters(x) is true if x glitters and predicate gold(x) is true if x is gold. What which one of the following logical formulate represents the above statement? Correct Answer ∃x: glitters (x) ∧ ¬ gold (x)

Concept:

¬ ∀x (P(x)) ≡ ∃ x ¬ P(x)

p → q ≡ ¬ p ∨ q

¬ (p ∨ q) ≡ ¬ p ∧ ¬ q

Explanation:

“All that glitters is gold” is expressed as

∀x (glitters (x) ⇒ gold (x))

“Not All that glitters is gold”

¬ ∀ x (glitters (x) ⇒ gold (x))

∃ x ¬ (glitters (x) ⇒ gold (x))

∃ x ¬ (¬ glitters (x) ∨ gold (x))

∃ x (glitters (x) ∧ ¬ gold (x))

Therefore option 4 is correct.

Important Point:

"Not all that glitters is gold" means "There exist a glitter and it is not gold" which can be expressed as

∃ x (glitters (x) ∧ ¬ gold (x))

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