Let Q(x, y) denote “x + y = 0” and let there be two quantifications given as…
2012
Let Q(x, y) denote “x + y = 0” and let there be two quantifications given as
(i) ∃y ∀x Q(x, y)
(ii) ∀x ∃y Q(x, y)
Where, x and y are real numbers. Then which of the following is valid?
- A.
I is true and II is false
- B.
I is false and II is true
- C.
I is false and II is also false
- D.
both I and II are true
Attempted by 46 students.
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Correct answer: B
First, analyze statement (i): ∃y ∀x Q(x, y). This requires a single fixed y that satisfies x + y = 0 for every real number x. Since y must equal -x, no single value works for all x, making this statement false. Next, analyze statement (ii): ∀x ∃y Q(x, y). For any chosen x, we can always find a corresponding y = -x to satisfy the equation. Thus, statement (ii) is true.
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