Discussion
Let \(\theta(x, y, z)\) be the statement "x + y = z" and let there be two quantifications given as
(i) \(\forall x\, \forall y\, \exists Z\ \theta(x, y, z)\)
(ii) \(\exists Z\, \forall x\, \forall y\ \theta(x, y, z)\)
Where x, y, z are real numbers. Then which one of the following is correct ?
Answer: B
Statement (i), for all x and y there is a z with x + y = z, is true, since for any real x and y the sum x + y is a real number z. Statement (ii), there is a z such that for all x and y, x + y = z, is false, because a single z cannot equal the sum of every pair, for instance x = y = 0 needs z = 0 while x = 1 and y = 0 needs z = 1. So (i) is true and (ii) is false, which is why the order of the quantifiers matters. The keyed answer is the second option.
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