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?
1 Answer
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".
