1964 AHSME Problems/Problem 12

Problem

Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$ ?

$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0\qquad \textbf{(E)}\ \text{For some x}, x^2 \le 0$

Solution

In general, the negation of a universal ("for all") quantifier will use an existential ("there exists") quantifier, and negate the statement inside.

In this case, we change the "For all" to "There exists", and negate the inner statement from $x^2 > 0$ to $x^2 \le 0$ .

So, the negation of the original statement is "There exists an $x$ such that $x^2 \le 0$ ". Exactly one of the two statements is true, but not both. This is the same as answer $\boxed{\textbf{(E)}}$ .

1964 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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1964 AHSME Problems/Problem 12

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