MathDB
Difficult Logic

Source: AHSME 1963 Problem 26

January 11, 2014
AMC

Problem Statement

Form 1 Consider the statements:
<spanclass=latexbold>(1)</span> p qr<spanclass=latexbold>(2)</span> p qr<spanclass=latexbold>(3)</span> p q r<spanclass=latexbold>(4)</span> p q r<span class='latex-bold'>(1)</span>\ p\text{ } \wedge\sim q\wedge r \qquad <span class='latex-bold'>(2)</span>\ \sim p\text{ } \wedge\sim q\wedge r\qquad <span class='latex-bold'>(3)</span>\ p\text{ } \wedge\sim q\text{ }\wedge \sim r \qquad <span class='latex-bold'>(4)</span>\ \sim p\text{ } \wedge q\text{ }\wedge r ,
where p,q,p,q, and rr are propositions. How many of these imply the truth of (pq)r(p\rightarrow q)\rightarrow r?

<spanclass=latexbold>(A)</span> 0<spanclass=latexbold>(B)</span> 1<spanclass=latexbold>(C)</span> 2<spanclass=latexbold>(D)</span> 3<spanclass=latexbold>(E)</span> 4<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1\qquad <span class='latex-bold'>(C)</span>\ 2 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 4

Form 2 Consider the statements (1)(1) pp and rr are true and qq is false (2)(2) rr is true and pp and qq are false (3)(3) pp is true and qq and rr are false (4)(4) qq and rr are true and pp is false. How many of these imply the truth of the statement "rr is implied by the statement that pp implies qq"?
<spanclass=latexbold>(A)</span> 0<spanclass=latexbold>(B)</span> 1<spanclass=latexbold>(C)</span> 2<spanclass=latexbold>(D)</span> 3<spanclass=latexbold>(E)</span> 4<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1\qquad <span class='latex-bold'>(C)</span>\ 2 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 4