I've been stuck on this question for some time, if anyone happens to solve it please explain step by step.
$$(A +B ) \times ( A' + C ) \times ( B + C )$$
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Please provide your efforts towards the solution. – shardulc Feb 25 '16 at 12:31
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What does $\times$ mean? Is it and operation? – Win Vineeth Feb 25 '16 at 12:34
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Multiplication and the ( ' ) is comploment – GhettoBurger Feb 25 '16 at 12:35
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@win-vineeth Yes, it is a quite standard notation. – J.-E. Pin Feb 25 '16 at 12:36
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@GhettoBurger You will have to wait. It's not like every person on the site is waiting for your question to answer it. It also helps if you provide the details of what you tried. We can know your level of understanding to help you out better. – Win Vineeth Feb 25 '16 at 12:39
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1Thanks for helping out everyone I'm new to stackoverflow but so far I'm really liking the community – GhettoBurger Feb 25 '16 at 13:03
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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 $a + a' = 1$.
J.-E. Pin
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I get that I have to use a law such as (complementary law) I did and the answer turned out to be AB + A'C. When I checked the answers the lecturer provided us with the answer was AC + A'B. – GhettoBurger Feb 25 '16 at 12:43
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@J.-E.Pin Re my deleted answer, I just wanted to work it out, not really give the game away. OP seemed totally befuddled, so I posted it. But I suppressed it, offered just a glimpse. – BrianO Feb 25 '16 at 12:49
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$(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$
=$A' \times B + A' \times B \times C+ A \times C + A \times B \times C$
=$A' \times B + A \times C$
Win Vineeth
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@J.-E.Pin Well, He says he tried. If I were him and I tried for an hour or so, I would really hate the site for not helping out. And, he did know all the laws... So, may be he made a silly mistake. Under these assumptions, I put the answer. – Win Vineeth Feb 25 '16 at 13:44
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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.
BrianO
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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): $X Y + \overline X Z + Y Z = X Y + \overline X Z$ $$AC + \overline AB$$
