Boolean Algebra: Simplifying multiple XOR and XNOR

Question

Is there any way to simplify a combination of XOR and XNOR gates in the following expression? I have tried multiple boolean theorems and I have not been able to simplify this any further:

The simplified version is XOR(A,B)*XNOR(C,D) + XOR(C,D)*XNOR(A,B)

The actual expression is: (A'B + AB')(CD + C'D') + (AB + A'B')(C'D + CD')

According to my lab, this expression can be simplified, but I don't know where to start.

I guess you could look at each element, such as (A'B + AB') as a single variable, such as X and then the equation would be XY' + X'Y, but I still don't see how to simplify that.

Answer 1

You're on the right track. Note that $X\bar Y + \bar X Y$ is exactly $X\oplus Y$ (where $\oplus$ is XOR), so you can simplify to $$ (A\oplus B) \oplus (C\oplus D)$$ Since $\oplus$ is associative and commutative, this simplifies further to $$ A\oplus B\oplus C\oplus D $$ which is particularly nice because it's a linear function of the inputs in the associated Boolean ring (whose multiplication is conjunction and addition is $\oplus$).

Answer 2

The definition (xor, xnor) of XOR(A,B)is A'B + AB' and of XNOR(C,D) is CD+C'D'. So XOR(A,B)*XNOR(C,D) + XOR(C,D)*XNOR(A,B) is a direct translation of (A'B + AB')(CD + C'D') + (AB + A'B')(C'D + CD').