1974 AHSME Problems/Problem 13
Problem
Which of the following is equivalent to "If P is true, then Q is false."?
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."
The fact that a statement's contrapositive is logically equivalent to it can easily be seen from a venn diagram arguement.
From this venn diagram, clearly "If , then ." is true. However, since is fully contained in , the statement "If not , then not ." is also true, and so a statement and its contrapositive are equivalent.
See Also
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Followed by Problem 14 | |
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