X Y Z+X Y' Z'+X' Y' Z+X' Y Z'

I know it simplify to (X XOR Y XOR Z),BUT I want to simplified using only AND, OR, and NOT Gates?

Please help I spent three hours but I don't get the same truth table.

  • $\begingroup$ If you want a sum of products, then $X Y Z+X Y' Z'+X' Y' Z+X' Y Z'$ is as simplified as you can get it. $\endgroup$ – Adriano Oct 4 '14 at 0:08
  • $\begingroup$ Thank you I though am going crazy. $\endgroup$ – Abdalla Oct 4 '14 at 0:29

I suppose the prime means negation and then $X'=1+X$ etc.

$XYZ+XY'Z'+X'Y'Z+X'YZ'=$ $XYZ+X(1+Y)(1+Z)+(1+X)(1+Y)Z+(1+X)Y(1+Z)=$ $XYZ+X(1+Y+Z+YZ)+(1+X+Y+XY)Z+(Y+XY)(1+Z)=$ $XYZ+X+XY+XZ+XYZ+Z+XZ+YZ+XYZ+Y+YZ+XY+XYZ=$ $X+Y+Z$

Since $X+Y=(X\vee Y)\wedge \neg(X\wedge Y)$ just use AND, OR and NOT to construct $X+Y+Z$, that is $X$ XOR ($Y$ XOR $Z$)


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