Boolean algebra rule?

Question

I am trying to simplify something easy, but it has been a while and ive forgotten some rules:

(a + b) * (~b + c) =? (a * ~b) + (a * c) + (b * ~b) + (b * c) =? [a * (~b + c)] + [b * (~b + c)] 

ok so far i have:  a(~b+c) + b(~b+c)
there is a rule for this but i cant remember do both the (~b+c) just cancel out like in regular math or what is the rule?

The final answer should be: (a + c)

I know there are other steps that be be taken to simplify differently, however I want to know what to do when your in this situation: a(~b+c) + b(~b+c)

? · Accepted Answer

In the previous step, you had:
(a * ~b) + (a * c) + (b * ~b) + (b * c) 

At this point, if you wanted, you could just remove the (b * ~b) because that will always be 0.  That would leave you with:
(a * ~b) + (a * c) + (b * c) 

(I am also pretty sure that you cannot simplify (a + b) * (~b + c) into just (a + c).  For example, if a = 0 and c = 1, then the result is 1 if b = 1, but the result is 0 if b = 0.  Since the result depends on b, there's no way you can just simplify it away.)

Boolean algebra rule?

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