1962 AHSME Problems/Problem 30


Consider the statements:

$\textbf{(1)}\ \text{p and q are both true}\qquad\textbf{(2)}\ \text{p is true and q is false}\qquad\textbf{(3)}\ \text{p is false and q is true}\qquad\textbf{(4)}\ \text{p is false and q is false.}$

How many of these imply the negative of the statement "p and q are both true?"

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$


By De Morgan's Law, the negation of $p$ and $q$ are both true is that at least one of them is false, with the exception of the statement 1, i.e.;

$\text{p and q are both true}.$

The other three statements state that at least one statement is false. So, 2, 3, and 4 work, yielding an answer of 3, or $\textbf{(D)}.$

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