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I am desperately trying to figure out the formal proof for this argument.

$\begin{array}{r} A\lor B\\ A\lor C\\ \hline A\lor (B \land C) \end{array}$

I am trying to apply the backwards method here. I am trying to infer A, in order to use vIntro in the last step and introduce the final disjunction. But I got stuck finding sufficient proof for A.

Any hint will be greatly appreciated. Thank you!

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    the OP keeps asking these kinds of (very basic) questions, as has been observed in another comment elsewhere. The problem is that he refuses to provide further information on his deduction method or book reference. He does not even bother to edit his own questions to provide extra information as requested by other members. It certainly looks like a lazy student asking us to do his homework. Prove me wrong (by the "backwards method").2011-12-16

4 Answers 4

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Is the following a "formal proof"?

Let $a$, $b$, $c$ be boolean variables representing the truth values of $A$, $B$, $C$. Then by the second distributive law of Boolean algebra we have

$(a\vee b)\wedge(a\vee c)=a\vee(b\wedge c)\ .$

This shows that your "argument" not only proves the truth of the third line under the assumption of the first two, but that in fact the stuff above the \hline is logically equivalent to what's underneath.

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Since you've pointed out the text elsewhere these problems seem to come from, here goes:

1 (A v B) premise 2 (A v C) premise 3 | A hypothesis 4 | (A v (B^C)) 3 V introduction 5 | B hypothesis 6 || A hypothesis 7 || (A v (B^C)) 6 V introduction 8 || C hypothesis 9 || (B^C) 5, 8 ^ introduction 10 || (Av(B^C)) 9 V introduction 11 | (Av(B^C)) 6-7, 8-10, 2 V elimination 12 (Av(B^C)) 3-4, 5-11, 1 V elimination.