A smal problem with boolean algebra

[gipc]

There's something I can't seem to figure out.

say we have the function
f=xz +x'z'

why is this equals (x xor z)' ?
i thought using De-Morgan's law we shall get

(x xor z)'=(x'z+z'x)'=(xz'zx')=0

then why does (x xor z)'=xz+x'z' ?


and one more small thing,
say we have
(x xor y) xor (x xor z)'
why does (x xor y) xor (x xor z)' = (x xor y xor x xor z)'

how did they come to this term?

 

[vela]

De Morgan's law also tells you (ab)' = a' + b', so

(x xor z)' = (x'z+z'x)' = (x'z)'(z'x)' = ...

I'll let you finish it.

 

[gipc]

thanks so much i feel so stupid for missing the obvious.

one more thing though,
why does the following equation stand?

(x XOR y) XOR (x XOR z)' = (x XOR y XOR x XOR z)'

 

[vela]

That's kind of like asking why a trig identity is true. It's true because it's a consequence of how things are defined. You just have to prove it by using algebraic manipulations or just listing all possible values of x, y, and z and showing the two sides give the same results.