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The (second) distributive law in boolean algebra is defined as

$A + (B C) = (A + B) (A + C)$

But wouldn't it be correct to define it that way:

$(A + (B C)\, ) = (A + B) (A + C)$

Because if you transform

$(\sim D + \sim C) (\sim D + B + A) (C + \sim B + \sim A)$

to

$\sim D + (\sim C (B + A) \, ) (C + \sim B + \sim A)$

it would be incorrect since * has higher precedence than +, so

$(\sim D + (\sim C (B + A)\, )\, ) (C + \sim B + \sim A)$

would be correct, or am I totally mistaken?

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    $\begingroup$ I don't understand your question. The second expression is identical to the first, except that it has a redundant pair of parentheses. $\endgroup$ – David Richerby Feb 18 '15 at 0:40
  • $\begingroup$ From my view the first one means not D or the rest so if D is low the output is high. The second one binds the D to the rest. $\endgroup$ – Benjoyo Feb 18 '15 at 0:47
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    $\begingroup$ I believe he means the very first two laws that you wrote. There is literally no difference because A+(BC)=(A+(BC)) because no matter the expression E=(E). This is a rule that goes far far beyond just boolean algebra and holds in every any algebra and logic I can think of $\endgroup$ – Jake Feb 18 '15 at 3:41
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The real problem is you're making a mistake with the precedence rules being used. The sentence: $$ (\lnot D + \lnot C)(\lnot D + B + A)(C + \lnot B + \lnot A) $$ is ambiguous until we add rules of precedence, or explicitly parenthesise it. Because logical $\mathsf{AND}$ is associative (and commutative), we normally ignore the fact that we have to decide, one way or the other, which of the two $\mathsf{AND}$s applies first.

The normal rules we use (when nothing else gets in the way, go left to right) implicitly parenthesises the sentence as: $$ [(\lnot D + \lnot C)(\lnot D + B + A)](C + \lnot B + \lnot A) $$

If we parenthesise in this explicit way, and apply the distributive law, then you get what you expect: $$ [\lnot D + \lnot C(B + A)](C + \lnot B + \lnot A) $$

If we changed our rules of precedence to right-to-left and thought about it as $$ (\lnot D + \lnot C)[(\lnot D + B + A)(C + \lnot B + \lnot A)] $$ then we can't apply the distributive law at all, we have to apply the associative property first, then we again get what you expect.

In general you have to remember that we often simplify mathematical notation to ease reading, so when you are manipulation expressions, you have to remember that we're leaving things out, or put them back in for clarity.

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