# How to prove $((Q \wedge ¬P) \vee (Q \wedge P)) = Q$

I cannot see any steps to this problem! Surely the answer is obvious? Is there a particular law which is used to make this statement?

$$((Q \wedge ¬P) \vee (Q \wedge P)) = Q$$

-
Shouldn't it be $$((Q \wedge ¬P) \vee (Q \wedge P)) = Q$$? – Peter Tamaroff May 7 '12 at 3:03

2
It's hard to say, because you haven't said what steps are allowed. For example, if you already know that $\vee$ distributes over $\wedge$, the thing is very simple: you factor out the $Q$ from the disjuncts, obtaining $(Q\wedge(P\vee\lnot P))=Q$, etc. – MJD May 7 '12 at 3:04

1
MarkDominus yes you are right, I can see the answer now. $$(Q \wedge (P \vee ¬P)) = Q$$ (Distributive Law) $$(Q \wedge T) = Q$$ (Excluded Middle) $$Q = Q$$ (Identity) – Danny Rancher May 7 '12 at 3:06

## 2 Answers

HINT: $$(Q\land\lnot P)\lor(Q\land P)\equiv Q\land(\lnot P\lor P)$$ by distributivity. Or of course you can simply use a truth table, if you're allowed to do so in this problem.

-

@Mark: I'm sure that I didn't. Thanks for the quick catch. – Brian M. Scott May 7 '12 at 3:07

The shape of this makes me think of DeMorgan's Laws. You could start with $Q=Q \vee False$ But you could just do a truth table.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 6, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9526646137237549, "perplexity_flag": "middle"}