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Boolean troubles help please

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I am reviewing for a test and there is a particular problem i am having trouble with. its solved but i dont see how or why certain steps are taken F = (A′ + BD′ + AD) (B + C′) (A+ B′C) step 1 very confusing to me (A'+BD'+D) * B(A+B'C) + C'(A+B'C) I cant see to figure out how that first step was accomplished.(and the rest of it is confusing but not as much) answer is F = ABD'+ ABD+ ABC'D'+ AC'D If someone could talk me through step by step I would appreciate it and hopefully learn something.

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(A′ + BD′ + AD) * (B + C′) * (A + B′C)
(A′ + BD′ + D) * (B + C′) * (A + B′C) // X + X'Y = X + Y
(A′ + B + D) * (B + C′) * (A + B′C) // X + X'Y = X + Y
(A′ + B + D) * (B(A + B′C) + C′(A + B′C)) // distribution

I think you had forgotten a parenthesis here...

(A′ + B + D) * (B * (A + B′C) + C′ * (A + B′C))
(A′ + B + D) * (BA + C′A) // XX' = false
(A′ + B + D) * (B + C′) * A
(B + D) * (B + C') * A // XX' = false
BA + DBA + BC'A + DC'A // distribution

This is close to your solution, but it can be shortened, since ABD' (AB by me) + ABD = AB

AB + AC'D

PS: If you say that it's matter of school no one will answer you questions!

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First off, is that really after the first step? I would think it would be more like:

(A'+BD'+AD) * (B(A+B'C) + C'(A+B'C))?

It sometimes helps to know the terms of algebra that are used to help simplify equations. In your case, you're going to be using repeated application of the distributive property. This says the following equality is true for certain types of number systems (in this case boolean number systems. But also matrices, the 'reals', and many others):

(B+C)A = BA + CA

This is the property that is used in the first step of your derivation. In this case, (A' + BD' + AD) is ignored, (A+B'C) is treated as one 'element', which we'll call E for now (because all the previous letters are taken. So, only paying attention to the relevant variables we get:

(B+C')E

which looks like the above property which gives us:

BE+C'E

Because E equals A+B'C we substitute E for it and get:

B(A+B'C) + C'(A+B'C)

But we 'ignored' (A'+BD'+AD) and we have to add it in:

(A'+BD'+AD)* ( B(A+B'C) + C'(A+B'C) )

I would just continue applying this until you get a series of what is called Sum of Products. Once you get that far you can utilize some additional properties of boolean numbers: A * A' = 0, A*A = A, etc.

Hope that helps a little.

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Quote:
Original post by guy128
I am reviewing for a test and there is a particular problem i am having trouble with. its solved but i dont see how or why certain steps are taken

F = (A′ + BD′ + AD) (B + C′) (A+ B′C)

step 1 very confusing to me
(A'+BD'+D) * B(A+B'C) + C'(A+B'C)

I cant see to figure out how that first step was accomplished.(and the rest of it is confusing but not as much)


Left side: A' implies A'D, so we have (with some redundancy) (A' + BD' + AD + A'D). AD + A'D is simply D (i.e. "D whether or not A").

Right two brackets: distribution. "(Either B or not C), and something interesting" is equivalent to "Either (B and something interesting) or (not C and something interesting)".

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I have a question that follows from reading this thread what the, is this dash notation the only time I've ever encountered such dashes in mathematics is to signify the order of a derivative.

for example f'(x) or f''(x)
but have never encountered A' or at least it has been wiped from my memory.

So what I'm asking is for some one to explain what area of mathematics this occurs in and if possible a short tutorial.

EDIT: just got an inkling thats it's notation used in matrix mathematics.

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Quote:
Original post by ramearess
I have a question that follows from reading this thread what the, is this dash notation the only time I've ever encountered such dashes in mathematics is to signify the order of a derivative.

for example f'(x) or f''(x)
but have never encountered A' or at least it has been wiped from my memory.

So what I'm asking is for some one to explain what area of mathematics this occurs in and if possible a short tutorial.

EDIT: just got an inkling thats it's notation used in matrix mathematics.


The area of mathematics is boolean logic (hence the thread's title). The dash signifies NOT (another common symbol for NOT is the tilde ~). The plus symbol signifies OR. The asterisk signifies AND. Also, two symbols next to each other signify AND.

So, AB' + C'D

Reads as:

(A and not(B)) or (not(C) and D)

Hope that helps,
Matt

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