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I have a solution to a logic problem involving propositions that I don't undersand how a particular step was carried out.The professor called the step I'm having trouble with reverse distribution.

Prove: $(p \lor q ) \land\lnot (p \lor \lnot q)\leftrightarrow p$

$(p \lor q ) \land (p \lor \lnot q) $

$p\lor(q \land \lnot q)$ This is the step I don't understand.

$p\lor FALSE $

p

The second step is throwing me for a loop. What am I not seeing?

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1 Answer 1

up vote 8 down vote accepted

Since $\vee$ distributes over $\wedge$, you know that $A\vee(B\wedge C)$ is equivalent to $(A\vee B)\wedge(A\vee C)$. So if you have the former, you can replace it with the latter.

But, likewise, if you have the latter, you can replace it with the former. Your second line, $(p\wedge q)\vee(p\wedge\neg q)$ is of the form $(A\vee B)\wedge(A\vee C)$ (with $A=p$, $B=q$, and $C=\neg q$), so it is equivalent to $A\vee(B\wedge C)$, which is exactly the third line.

In other words: instead of using the "distributive property" as usual, you use it "in reverse". It's much like going from $5\times 3 + 5\times 7$ to $5\times(3+7)$, instead of the other way around. You can think of it as the analogue of "factoring out" instead of "distributing through".

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Dr. Magidin I really appreciate all the answers over the last few weeks. you have been very help. Thank you. –  lampShade Jan 31 '11 at 4:02
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