Simplify the boolean function below by using algebra laws. I've been stuck on this question for some time, if anyone happens to solve it please explain step by step. 
$$(A +B ) \times ( A' + C ) \times ( B + C )$$
 A: Since you are stuck, I will just indicate the first steps as hints
Step 1. Use distributivity to get a sum of products (instead of products of sums).
Step 2. Simplify using rules like $a + a = a$, $aa' = 0$  and $a + a' = 1$.
A: $(A +B ) \times ( A' + C ) \times ( B + C ) = [(A+B)\times A' + (A+B)\times C]\times (B+C) $
=$[A\times A'+B \times A' + A \times C + B \times C]\times (B+C) $
$A \times A' =0$
So, $B \times B \times A' +B \times A' \times C + A \times B \times C +A \times C \times C + B \times B \times C + B \times C \times C$
=$A' \times B + A' \times B \times C+ A \times B \times C+A \times C +B \times C = A' \times B + A \times C + B \times C$
write $B \times C$ as $A \times B \times C + A' \times B \times C$
=$ A' \times B + A \times C + A \times B \times C + A' \times B \times C$
=$A' \times B  + A' \times B \times C+ A \times C + A \times B \times C$
=$A' \times B + A \times C$
A: Multiply it out to get a "flat" expression that's a sum of 8 terms $XYZ$, where each $X,Y,Z$ is one of $A,A',B,C$, possibly with more than one occurrence.
Then simplify, simplify, simplify. Don't forget the absorption law: $XY+Y = Y$ (and so $XYZ+XY = XY$). The whole thing reduces to something much simpler.
A: You need to dig out your textbook and do some work.  There are lots of tricks, which only come from doing the work.
$$(A+B)×(\overline A+C)×(B+C)$$
$$(A+B)\ (\overline A+C)\ (B+C)$$
Less busy if you remove $×$.  Think of it like addition and multiplication.
Consensus Law (13a): $(X + Y)\ (\overline X + Z)\ (Y + Z) = (X + Y)\ (X + Z)$
$$(A+B)\ (\overline A+C)$$
Distributive Law (8b): $(W + X)\ (Y + Z) = W Y + W Z + X Y + X Z$
$${A \overline A} + AC + \overline AB + CB$$
Complement Law (4a): $X • \overline X = 0$
$$AC + \overline AB + CB$$
Consensus Law (13b): $X Y + \overline X Z + Y Z = X Y + \overline X Z$
$$AC + \overline AB$$
Laws and Theorems of Boolean Algebra
