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I have the standard logical equivalence: $(p\rightarrow q)\wedge(q\rightarrow r)\Leftrightarrow p\rightarrow (q\wedge r)$.

Using several distributive laws I was able to get it down to: $(\neg p\wedge\neg q) \vee (\neg p\wedge r) \vee (q\wedge r)$.

I must be missing some manipulation I can do to reduce this.

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    on second thought, for equlivalance your question should be $[(p\Rightarrow q)\wedge(q\Rightarrow r)\Leftrightarrow p]\Rightarrow (q\wedge r)$2013-09-08

2 Answers 2

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Umm... maybe I am missing something, but

if $p$ is false, $q$ is true and $r$ is false, then we have that

$(p\rightarrow q)\wedge(q\rightarrow r)$ is false

$p\rightarrow (q\wedge r)$ is true.

So I don't see how you can prove the equivalence.

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$\begin{array}{|c|c|c|c|c|c|c|c|c}\hline p&q&r&p\Rightarrow q&q\Rightarrow r&(p\Rightarrow q)\land (q\Rightarrow r)&q\land r&p\Rightarrow(q\land r)&\cdots\\\hline T&T&T&T&T&T&T&T\\T&T&F&T&F&F&F&F\\T&F&T&F&T&F&F&F\\T&F&F&F&T&F&F&F\\F&T&T&T&T&T&T&T\\F&T&F&T&F&F&F&T\\F&F&T&T&T&T&F&T\\F&F&F&T&T&T&F&T\\\hline \end{array}$

$ \begin{array}{c|c|c|}\hline \cdots&\big[(p\Rightarrow q)\land (q\Rightarrow r)\big]\iff \big[ p\Rightarrow(q\land r)\big]\\\hline &T\\&T\\&T\\&T\\&F\\&T\\&T\\\hline \end{array}$