# 1958 AHSME Problems/Problem 44

## Problem

Given the true statements: (1) If $a$ is greater than $b$, then $c$ is greater than $d$ (2) If $c$ is less than $d$, then $e$ is greater than $f$. A valid conclusion is: $\textbf{(A)}\ \text{If }{a}\text{ is less than }{b}\text{, then }{e}\text{ is greater than }{f}\qquad \\ \textbf{(B)}\ \text{If }{e}\text{ is greater than }{f}\text{, then }{a}\text{ is less than }{b}\qquad \\ \textbf{(C)}\ \text{If }{e}\text{ is less than }{f}\text{, then }{a}\text{ is greater than }{b}\qquad \\ \textbf{(D)}\ \text{If }{a}\text{ is greater than }{b}\text{, then }{e}\text{ is less than }{f}\qquad \\ \textbf{(E)}\ \text{none of these}$

## Solution

(A) is not valid because $a does not imply $c\le d$ (The inverse of Statement 1 is not necessarily true).

(B) is not valid because $e>f$ does not imply $c (The converse of Statement 2 is not necessarily true either).

In (C), $e does imply $c\ge d$ (contrapositive of Statement 2) but this does not tell us anything about $a$ and $b$.

In (D), $a>b$ does imply $c>d$, but the inverse of Statement 2 ( $e\le f$) is not necessarily true.

So the answer is $\boxed{\textbf{(E) } \text{None of these}}$. (Incidentally, the wording of this problem appears to be wrongly assuming that $x\not implies $x>y$ (and vice versa) when in reality $x$ and $y$ could be equal. However, this does not change the fact that none of the four choices follow logically from the given statements.)

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