How would I be able to reduce this boolean expression? $(b + d)(a' + b' + c)$

Question

So, I have this boolean expression and I have to simplify it, here is what I am doing:

$(b + d) * (a' + b' + c)$

*Opening the expression by multiplication

$= a'b + bb' + bc + a'd + b'd + cd$

$= a'b + bc + a'd + b'd + cd$

But when I go and check the answer using an online simplifier, the answer is:

ANS = $a'b + bc + b'd$

But I can't think of any steps to reduce my equation to the required answer.

Any help would be appreciated.

Answer 1

You can use the Consensus Theorem:

$$xy + x'z + yz = xy + x'z$$

Applied to what you have, you can eliminate the $a'd$ term since you have both $a'b$ and $b'd$, and you can also eliminate the $cd$ term that is the 'consensus' of $bc$ and $b'd$

If you want to do all this with more elementary equivalences, please know that the Consensus Theorem itself is easily derived:

$$xy + x'z + yz \overset{Adjacency}{=} $$

$$xy + x'z + xyz + x'yz \overset{Absorption \times 2}{=}$$

$$xy + x'z$$

Applied to your expression, we can thus get:

$$a'b+bc+a'd+b'd+cd \overset{Adjacency \times 2}{=} $$

$$a'b+bc+a'bd+a'b'd+b'd+cbd + cb'd\overset{Absorption \times 4}{=}$$

$$a'b+bc+b'd$$