1963 AHSME Problems/Problem 26
Problem
Consider the statements:
where , and are propositions. How many of these imply the truth of ?
Solution
Statement states that is true and is false. Therefore, is false, because a premise being true and a conclusion being false is, itself, false. This means that , where is any logical statement (or series of logical statements) must be true - if your premise is false, then the implication is automatically true. So statement implies the truth of the given statement.
Statement similarly has as true and is false, so it also implies the truth of the given statement.
Statement states that and are both false. This in turn means that is true. Since is also true from statement , this means that is true, since is . Thus statement implies the truth of the given statement.
Statement states that is false and is true. In this case, is true - your conclusion can be true even if your premise is false. And, since is also true from statement , this means is true. Thus, statement implies the truth of the given statement.
All four statements imply the truth of the given statement, so the answer is
See Also
1963 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 25 |
Followed by Problem 27 | |
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