Let p,q,r,s represent the following propositions. p: \(x \in \{8, 9, 10, 11,…
2016
Let p,q,r,s represent the following propositions.
p: \(x \in \{8, 9, 10, 11, 12\}\)
q: \(x\) is a composite number
r: \(x\) is a perfect square
s: \(x\) is a prime number
The integer \(x\) ≥ 2 which satisfies \(¬((p ⇒ q)∧(¬r∨ ¬s))\) is _____________ .
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Correct answer: 11
Key idea: simplify the logical expression and test numbers in the given set.
Interpret the predicates: p: x ∈ {8, 9, 10, 11, 12}; q: x is a composite number; r: x is a perfect square; s: x is a prime number.
Simplify the formula: ¬((p ⇒ q) ∧ (¬r ∨ ¬s)) ≡ ¬(p ⇒ q) ∨ ¬(¬r ∨ ¬s) ≡ (p ∧ ¬q) ∨ (r ∧ s).
Note r ∧ s is impossible for x ≥ 2: a perfect square n = k² with k ≥ 2 is composite, and k = 1 gives n = 1 which is below the domain, so r ∧ s is false.
Thus we need p ∧ ¬q: x must be in {8,9,10,11,12} and not composite (i.e., prime). Checking the set, only 11 is prime.
Answer: 11
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