# Talk:Boole's expansion theorem

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Field:  Discrete mathematics

## Names

Two articles with similar names, both quite confusingly written for new readers. Would be good to expand them with examples/combine into one. I would, but came here looking for info on Shannon, so I'll update it when I've figured out what to write. Bwgames 15:13, 22 January 2006 (UTC)

I've added info based on a copy of Shannon's 1948 seminal paper that I have with me. Help appreciated with further editing (including cleaning the example and merging in the page "Shannon's expansion theorem", which is misnamed - this is not a theorem). 213.58.128.215 13:38, 19 July 2007 (UTC) A.B.Leal

## Copyright and Merger Fixed

I merged the files together but can't delete the other one - the other one was kind of copyrighted but now that has been resolved - I fixed the mess. Hope it makes a lot of sense now! LOTRrules (talk) 14:50, 27 January 2008 (UTC)

## Citation

Does anyone has a "real" citation for the proof of Bool? I would be glad if one could add one or -- if not -- change the sentence into smth like "is normally attributed to George Boole". thanks —Preceding unsigned comment added by 147.210.129.49 (talk) 13:58, 5 February 2008 (UTC)

This article was copied from this page when it was merged with Shannon's expansion theorem in 2008, the later one being a pure copy of the external page, so I put back the article before the merge. Freewol (talk) 14:27, 8 April 2013 (UTC)

## Improvements

This article needs work to be intelligible to beginners. Consider replacing the lede paragraph with this, if it's correct:

Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity
${\displaystyle F=x\cdot F_{x}+x'\cdot F_{x}'}$,
where ${\displaystyle F}$ is any Boolean function, ${\displaystyle x'}$ denotes the complement (negation) of ${\displaystyle x}$, and ${\displaystyle F_{x}}$and ${\displaystyle F_{x}'}$ are ${\displaystyle F}$ with the argument ${\displaystyle x}$ equal to ${\displaystyle 1}$ and to ${\displaystyle 0}$ respectively.

I also think more needs to be said about ${\displaystyle x}$ being a vector of Boolean values. Jess (talk) 21:06, 14 July 2016 (UTC)

Take a look at my edit. x is not a vector of Boolean values, but a Boolean-valued variable. --Macrakis (talk) 22:00, 14 July 2016 (UTC)

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## Proof

Can someone help flesh out the claim that "[the] Proof for the statement follows from direct use of mathematical induction, from the observation that ${\displaystyle f(X_{1})=X_{1}.f(1)+X_{1}'.f(0)}$ and expanding a 2-ary and n-ary Boolean functions identically." The base case for 1 variable is self evident, but the casual mention of expansion of 2-ary and n-ary functions seems unfounded. What am I missing? EulerPie (talk) 23:43, 25 September 2017 (UTC)