GATE CS 2016 Set 1 — Question 11
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Engineering Mathematics → Mathematical Logic → Propositional Logic
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Question
Let represent the following propositions.
is a composite number
is a perfect square
is a prime numberThe integer which satisfies is ________.
is a composite number
is a perfect square
is a prime numberThe integer which satisfies is ________.
Correct answer
11 to 11
Solution
We are given the logical expression: Using De Morgan's laws and logical equivalences:
So we need to find an integer that satisfies either or .Case 1: is true.
So we need to find an integer that satisfies either or .Case 1: is true.
- is true:
- is true: is NOT a composite number. Since , this means is prime.
- Checking the set from : (composite), (composite), (composite), (prime), (composite).
- The only value satisfying both is .
- is true: is a perfect square.
- is true: is a prime number.
- No integer is both a perfect square and a prime number. A perfect square has factors , so it has at least 3 factors (if ) or 1 factor (if ), making it composite or unit, never prime.
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