boolean algebra simplification for x(1 +bc') + x'(b' + bc)

Question

in this equation using boonlean algebra: X(1 +BC') + X'(B' + BC).

can i simplify (1 +BC') = 1 and (B' + BC) = B' +C?

i used truth table and they have the same result, but i do not know how to solve it using the rules?
thanks for all the help.

Answer 1

Yes. The first identity is due to Domination Law. The second is because: \begin{align*} B' + BC &= B'(1) + BC & \text{by Identity Law} \\ &= B'(1 + C) + BC & \text{by Domination Law} \\ &= B'(1) + B'C + BC & \text{by Distributive Law} \\ &= B'(1) + (B' + B)C & \text{by Distributive Law} \\ &= B'(1) + (1)C & \text{by Inverse Law} \\ &= B' + C & \text{by Identity Law} \\ \end{align*}