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Expert: - 2/12/2009


Question
I need to find the complements of the following functions
but I can't seem to get the right answer... >.< If you
could explain how to do these that would be great.

+ are ors
* are ands(Also a*b = ab
To show not I will use ~

First one is: Find the complement of F = wx + yz; then
show that FF~ = 0 and F + F~ = 1

What I got for the complement was
F = wx + yz = (w + x)(y + z) = (x~x~)(y~z~)
I couldn't figure out how to show the second part of the
problem even if I got the complement correct.

The next problem is. Find the Complement of the following
expressions.
xy~+x~y

I got this which I think is incorrect.
xy~ + x~y = (x+y~)(x~+y) = (x~+y)(x+y~) =
(x~y)(xy~)

The last one I am having problems with is this.

Implement the Boolean Function.
F = xy + x~y~ + yz~

a)with AND, OR, and inverter gates
b)with OR and inverter gates

I have NO idea where to begin with this one.

Thanks for the input/help!  

Answer
I will use C() as the complement, u as union and n as
intersection, and S for the entire space.  I believe
that u is an or, for which you have +, and that * is
an and.

There are three basic facts to know:
[1] C(A) = S - A,
[2] C(AuB) = C(A) n C(B),
[3] C(AnB) = C(A) u C(B).

For w, x, y, and z I will use W, X, Y, and Z (captialize them,
so that u and n appear as union and intersection.

What we have is F = WnX u YnZ.
To find C(F), it is C(WnX u YnZ).

By [3], the complement of a union is known to be the intersection
of the complement of each one, so C(WnX u YnZ) = C(WnX) n C(YnZ).

By [2], we know that C(WnX) = C(W) u C(X) and that
C(YnZ) = C(Y) u C(Z).

Sp the final result is [C(W) u C(X)] n [C(Y) u C(Z)].

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Now I will attempt to translate this into your terms.
-----------------------------------------------------------------

There are three basic facts to know:
[1] ~A = S - A,
[2] ~(A+B) = ~A * ~B,
[3] ~(A*B) = ~A + ~B.

For w, x, y, and z I will use W, X, Y, and Z (captialize them,
so that u and n appear as union and intersection.

What we have is F = W*X + Y*Z.
To find ~F, it is ~(W*X + Y*Z).

By [3], the complement of a union is known to be the intersection
of the complement of each one, so ~(W*X + Y*Z) = ~(W*X) * ~(Y*Z).

By [2], we know that ~(WnX) = ~W + ~X and that ~(Y*Z) = ~Y + ~Z.

So the final result is (~W + ~X) * (~Y + ~Z).

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Now I will do (a) and (b).
-----------------------------------------------------------------
(a) Take our answer, (~W + ~X) * (~Y + ~Z), and convert it to
using AND.  Use [3] to say that it is W*X * Y*Z .

(b) Take our answer, (~W + ~X) * (~Y + ~Z), and convert it to
using OR.  Using [2], we can say that it is
~[(~W + ~X) + (~Y + ~Z)].

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