Boolean Algebra: Simplifying product of sums

Question

I'm trying to simplify

 (A+B+C)(A+notB+C)(notA+B+notC)

The K-map gives me

(A+C)(notA+B+notC) 

but when I use boolean algebra I get

(A+AC+C)(notA+B+notC) 

Can someone explain to me how to simplify this using algebra?

Answer 1

Notice that: $$ A + AC + C = (A + AC) + C = A(1 + C) + C = A(1) + C = A + C $$

Answer 2

We'll focus on the first two factors. By the distributive law, we have $$(\color{red}{A}+\color{blue}B+\color{red}{C})(\color {red} A+\color{blue}{\lnot B} + \color{red}C) = [\color{red}{(A+C)}+\underbrace{(\color{blue}{B\cdot\lnot B)}}_{\large = 0}] = (A+C)$$

The second factor remains unchanged, to get $$(A+C)(\lnot A + B + \lnot C)$$