How to simplify boolean expressions.

Question

I'm struggling to understand what rules to apply when simplifying boolean expression.

For example:

$$ B+(A\cdot(C+B) \overline C) $$

I'm not sure how to simplify this expression. Here is my attempt.

$$ = B+AB\cdot(C+\overline C) \\ = B+AB\cdot(1) \\ = B(A+B) \\ = B(B+A) \\ = B $$

Answer 1

You can simplify this almost the same way you'd simplify an ordinary algebra expression:

$B+(A\cdot(C+B) \overline C)$

So just simplify it like an ordinary algebra expression first (technically we're using that AND is distributive over OR... in the same way multiplication is distributive over addition in ordinary algebra)

$=B+A(C\overline C+B\overline C)$. So here I'm just using $(C+B)\overline C = C\overline C + B\overline C$

Now I can use $C \overline C = 0$

$=B+A(0+B\overline C)$

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

$=B+B(A\overline C)$ (using commutativity and associativity of AND... just like in ordinary multiplication)

$=B(1+A\overline C)$ (factoring)

In boolean algebra 1+ anything is just 1.

$=B(1)$

$=B$

There are other tricks... but the initial main "tactic" is just to treat AND like multiplication and OR like addition.

Answer 2

Two really useful principles are:

Absorption

$P + PQ = P$

Reduction

$P+ P'Q = P + Q$

With these:

$ B+A(C+B)C' \overset{Reduction}{=}$

$B+ ABC' \overset{Absorption}{=}$

$B$

Done!