Which one of the following options is CORRECT given three positive integers…
2011
Which one of the following options is CORRECT given three positive integers \(x,y\) and \(z\), and a predicate
\(P\left(x\right) = \neg \left(x=1\right)\wedge \forall y \left(\exists z\left(x=y*z\right) \Rightarrow \left(y=x\right) \vee \left(y=1\right) \right)\)
- A.
\(P(x)\)being true means that\(x\)is a prime number - B.
\(P(x)\)being true means that\(x\)is a number other than - C.
\(P(x)\)is always true irrespective of the value of\(x\) - D.
\(P(x)\)being true means that\(x\)has exactly two factors other than 1 and\(x\)
Attempted by 72 students.
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Correct answer: A
Key idea: interpret the predicate P(x) in plain language.
The predicate is ¬(x = 1) ∧ ∀y ( (∃z (x = y*z)) ⇒ (y = x ∨ y = 1) ).
¬(x = 1) means x is not 1, and since x is a positive integer this means x > 1.
∀y ( (∃z (x = y*z)) ⇒ (y = x ∨ y = 1) ) says: every y that divides x must be either 1 or x; in other words, x has no divisors other than 1 and itself.
Combining these points, P(x) holds exactly for positive integers greater than 1 that have no divisors other than 1 and themselves — that is the definition of a prime number.
Examples and counterexamples:
x = 7 satisfies P(x): 7 ≠ 1 and its only positive divisors are 1 and 7.
x = 1 does not satisfy P(x) because ¬(x = 1) fails.
x = 4 does not satisfy P(x) because 2 divides 4 and 2 is neither 1 nor 4.
Conclusion: The correct interpretation of P(x) is that x is a prime number.
Note: Any statement claiming that P(x) means x has extra factors other than 1 and x, or that P(x) holds for all x, is false.
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