Explanation needed on rules for boolean simplification

Question

Good day

I am getting confused on Boolean function simplification. I dont quite understand how the equation is simplified as in the rules for simplification in boolean algebra.

Can anyone help explain how i would simplify these two examples.

AB'C + (AB'C)D'

x + (x + xy) + y(xy')'

Thank you

Answer 1

Here is a good reference http://www.electronics-tutorials.ws/boolean/bool_6.html So the first expression can be simplified to AB'C (if it's value is true, the whole expression is true, if it's false, the whole expression is false, D does not matter). The second expression can be reduced to x + y, I think (x+xy=x, x+x=x, etc).

Answer 2

Here are some basic simplification rules:

Double Negation

$P''=P$

DeMorgan

$(PQ)' = P' + Q'$

$(P + Q)' = P'Q'$

Absorption

$P + PQ = P$

$P(P+Q)=p$

Idempotence

$P+P=P$

$PP = P$

With these, you can 'solve' both of your problems.