so I have S = x\'y\'z + x\'yz\' + xy\'z\' + xyz I simplified it to S = (x\'y XOR

Question

so I have S = x'y'z + x'yz' + xy'z' + xyz

I simplified it to S = (x'y XOR x'z) + (xy' XOR xz'). I don't even know if it make sense to simplify... But the diagram of this can be seen in photo.

Then I have C = x'yz + xy'z = xyz' + xyz

I simplified that to C = (xy XOR xz) + yz, again the photo for this can be seen in the picture. Can anyone tell me if my diagrams are actually correct? Or if I should just not simplify these at all and just make a circuit diagram with the original equations?

uhhhh it is a wee bit sideways I guess :P

C = (x + y + z) (x + y + ) (x + y + z) ( + y + z) S = + y + x + xy ( + y ) + x( + yz) S = ( ) + x( ) S = ( + ) C = y + x + xy + xy rightarrow x( + y ) + y ( ) C = x(y ) + y rightarrow C = (xy x) + y

Explanation / Answer

Background: If constrction of a circuit is possible using XOR or XNOR gates it is prefferable beacuse AND,OR gates are more costly.

A simpler version:
S =x'y'z + xyz + xy'z' + x'yz'
= z(x'y' + xy) + z'(xy' + x'y)
= z(x XNOR y) + z'(x XOR y)
= z XOR (x XOR y) (This if you recall is the SUM function of a full adder)

For the 2nd one this seems more appropriate:
C = x.(yz' +y'z) + yz.(x+x') = x.(y XOR z) + y.z
This expression is infact the CARRY gate of a full adder where you get 1 when there are more than 2 1s. The expression you wrote is infact used to build a FULL ADDER using a half adder, where the the sum

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