1964 AHSME Problems/Problem 12

Revision as of 17:13, 23 July 2019 by Talkinaway (talk | contribs) (Created page with "== Problem== Which of the following is the negation of the statement: For all <math>x</math> of a certain set, <math>x^2>0</math>? <math>\textbf{(A)}\ \text{For all x}, x^2...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 $\boxed{\textbf{(E)}}$.

See Also

1964 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS