GATE CS 2014 Set 1 — Question 11
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Engineering Mathematics → Mathematical Logic → Predicate Logic & Quantifiers
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Question
Consider the statement"Not all that glitters is gold"Predicate is true if glitters and predicate is true if is gold. Which one of the following logical formulae represents the above statement?
Correct answer
(D) ∃ x: glitters(x) ∧ ¬ gold(x)
Solution
To represent the statement "Not all that glitters is gold" in first-order logic, we can break down its meaning step-by-step:
1.Analyze the natural language statement:
"Not all that glitters is gold" means "It is not the case that everything which glitters is gold."2.Translate "All that glitters is gold" into logic:
"Everything that glitters is gold" is represented as:3.Negate the statement:
"Not all that glitters is gold" is the negation of the above formula:4.Simplify using logical equivalences:
- Using the quantifier negation rule, :
- Using the implication equivalence, :
- Applying De Morgan's Laws to the negation:
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