1963 AMC 12 Problem 26

Question

Form I. Consider the statements

\begin{aligned} (1)\quad&p\land\neg q\land r,\\ (2)\quad&\neg p\land\neg q\land r,\\ (3)\quad&p\land\neg q\land\neg r,\\ (4)\quad&\neg p\land q\land r, \end{aligned}

where

p,

q,

r

are propositions. How many of these imply the truth of

(p\to q)\to r?

Form II. Consider the statements

(1)

p

and

r

are true and

q

is false,

(2)

r

is true and

p

and

q

are false,

(3)

p

is true and

q

and

r

are false,

(4)

q

and

r

are true and

p

is false. How many of these imply the truth of the statement “

r

is implied by the statement that

p

implies

q

”?

0

1

2

3

4

An implication is false only when its antecedent is true and its consequent is false

Evaluate

p\to q

first in each of the four assignments

In cases

(1)

and

(3),

p\to q

is false, so the outer implication is true. In cases

(2)

and

(4),

p\to q

is true, but

r

is also true, so the outer implication is again true. All four statements imply it.

Thus, the correct answer is E.

Problem 26 in other years