1974 AHSME Problems/Problem 13

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Problem

Which of the following is equivalent to "If P is true, then Q is false."?

$\mathrm{(A)\ } \text{P is true or Q is false."} \qquad$

$\mathrm{(B) \ }\text{If Q is false then P is true."} \qquad$

$\mathrm{(C) \ } \text{If P is false then Q is true."} \qquad$

$\mathrm{(D) \ } \text{If Q is true then P is false."} \qquad$

$\mathrm{(E) \ }\text{If Q is true then P is true."} \qquad$

Solution

Remember that a statement is logically equivalent to its contrapositive, which is formed by first negating the hypothesis and conclusion and then switching them. In this case, the contrapositive of "If P is true, then Q is false." is "If Q is true, then P is false." $\boxed{\text{D}}$

The fact that a statement's contrapositive is logically equivalent to it can easily be seen from a venn diagram arguement.

$[asy] draw(circle((0,0),3)); draw(circle((-1,1),1)); label("q",(1,-1)); label("p",(-1,1)); [/asy]$

From this venn diagram, clearly "If $p$, then $q$." is true. However, since $p$ is fully contained in $q$, the statement "If not $q$, then not $p$." is also true, and so a statement and its contrapositive are equivalent.