Is it possible to simplify this boolean expression?

Question

It threw me off because the variables are all barred.

$A' \downarrow B' ~\vee~ C' \downarrow D' ~~aka~~ \overline{\overline{A}\vee \overline{ B}}\vee\overline{\overline{C}\vee\overline{D}}$

Answer 1

$AB$ is "$A$ and $B$", $'$ means "not", $+$ means or, hence $A'$ nor $B'$ is $(A' + B')'$. So the starting value is $(A' + B')' + (C' + D')'$. We can simplify this:

$$\begin{equation} \begin{split} (A' + B')' + (C' + D')' & = A''B'' + C''D'' \quad (\textrm{De Morgan's Law})\\= AB+CD \qquad (A''=A) \end{split}\end{equation} $$ Hence, the expression can be rewritten as "($A$ and $B$) or ($C$ and $D$)". This is the simplification of your expression.