The statement P(\(x\)):"x = x2". If the universe of disclosure of disclosure…
2023
The statement P(\(x\)):"x = x2". If the universe of disclosure of disclosure consists of integers, what are the following have truth values :
(A) P(0)
(B) P(1)
(C) P(2)
(D) \(\exists x \ P(x)\)
(E) \(\forall x \ P(x)\)
Choose the correct answer from the options given below :
- A.
(A), (B) and (E) Only
- B.
(A), (B) and (C) Only
- C.
(A), (B) and (D) Only
- D.
(B), (C) and (D) Only
Attempted by 50 students.
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Correct answer: C
We are given P(x): x = x2 (interpreted as x = x^2) over the integers.
Key idea: Solve the equation x = x^2 to find which integers satisfy P(x).
Step 1: Rearranging gives x^2 - x = 0, i.e. x(x - 1) = 0.
Step 2: The integer solutions are x = 0 and x = 1.
Therefore evaluate each statement:
P(0): true (0 = 0^2).
P(1): true (1 = 1^2).
P(2): false (2 ≠ 2^2).
There exists x such that P(x): true (for example x = 0).
For all x P(x): false (not all integers satisfy the equation).
Conclusion: The true statements are P(0), P(1), and the existential statement. The correct selection lists P(0), P(1), and ∃x P(x).