question about the notation in boolean algebra?

Question

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 $A (B + C)$?

Answer 1

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.

Answer 2

Hint: Look up De Morgan's laws.