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Is there a difference between

  1. $\bar{A}\bar{B}$
  2. $\overline{AB}$

Is there a difference between

  1. $\bar{A}+\bar{B}$
  2. $\overline{A+B}$

Also, just to be sure, the equal sign is a normal equal sign in boolean algebra right? So the left and right can be reversed? For example I always see the Distributive Law written as

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

but if an equation has $A B + A C$ does this mean I can replace it with $A (B + C)$?

leftandright
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2 Answers2

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Yes, there is a difference. Think about the case if $B = \bar A$ then $\bar A \bar B = \bar A \bar {\bar A} = \bar A A = 0$, however $\overline{A B} = \overline{A\bar A} = \bar 0 = 1$. Thus $\overline{AB}\neq \bar A\bar B$.

The same thing is the case for you second question if we choose $B = \bar A$. $\bar A + \bar B = \bar A + \bar{\bar A} = \bar A + A = 1$, however $\overline{A+ B} = \overline{A + \bar A}= \bar 1 = 0$.

Equality however works just like normal equality.

Ove Ahlman
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  • Is there any way brackets can be used to make things more clear? For example $\bar{A}\bar{B}=(\bar{A})(\bar{B})$? – leftandright Sep 14 '15 at 08:07
  • What do you mean "more clear"? Since the expressions we look at only contain two symbols, brackets does not really make a big difference. Brackets are usefull when considering many symbols in order to see which two are supposed to be applied together first. Such as $A(B+C)$ where B+C is supposed to be done first, and then left multiplication with A. – Ove Ahlman Sep 14 '15 at 08:13
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Hint: Look up De Morgan's laws.

Jose Arnaldo Bebita Dris
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