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1963 AHSME Problems/Problem 26

Problem

Consider the statements:

$\textbf{(1)}\ p\text{ }\wedge\sim q\wedge r\qquad\textbf{(2)}\ \sim p\text{ }\wedge\sim q\wedge r\qquad\textbf{(3)}\ p\text{ }\wedge\sim q\text{ }\wedge\sim r\qquad\textbf{(4)}\ \sim p\text{ }\wedge q\text{ }\wedge r$

where $p,q$, and $r$ are propositions. How many of these imply the truth of $(p\rightarrow q)\rightarrow r$?

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

Solution

Statement $1$ states that $p$ is true and $q$ is false. Therefore, $p \rightarrow q$ is false, because a premise being true and a conclusion being false is, itself, false. This means that $(p \rightarrow q) \rightarrow X$, where $X$ 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 $1$ implies the truth of the given statement.

Statement $3$ similarly has $p$ as true and $q$ is false, so it also implies the truth of the given statement.

Statement $2$ states that $p$ and $q$ are both false. This in turn means that $p \rightarrow q$ is true. Since $r$ is also true from statement $2$, this means that $(p \rightarrow q) \rightarrow r$ is true, since $T \rightarrow T$ is $T$. Thus statement $2$ implies the truth of the given statement.

Statement $4$ states that $p$ is false and $q$ is true. In this case, $p \rightarrow q$ is true - your conclusion can be true even if your premise is false. And, since $r$ is also true from statement $4$, this means $(p \rightarrow q) \rightarrow r$ is true. Thus, statement $4$ implies the truth of the given statement.

All four statements imply the truth of the given statement, so the answer is $\boxed{\textbf{(E)}}$

See Also

1963 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 25 		Followed by
Problem 27
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

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