I need to simplify the following boolean expression
¬(A xor B) xor (B + ¬C)
I know A xor B = ¬AB + A¬B
Then the expression will become
¬(¬AB + A¬B) xor (B + ¬C)
However, I stuck on it and I don't know how to simplify it further. Can someone give me a hint or push me in the right direction?
Best Answer
If your expression is $¬(A \oplus B) \oplus (B \vee ¬C)$, then
$¬(A \oplus B)=1\oplus A\oplus B$, $¬C=1\oplus C$ and $B\vee\neg C=B\oplus(\neg C)\oplus B(\neg C)$ so
$¬(A \oplus B) \oplus (B \vee ¬C)=1\oplus A\oplus B\oplus B\oplus 1\oplus C\oplus B(1+C)=$ $A\oplus B\oplus C\oplus BC=$
Since $\oplus$ (XOR) is associative, and distributive over $\cdot$ (AND), and $X\oplus X=0$.
($0$ is FALSE and $1$ is TRUE)
The elements $\displaystyle\bigoplus_{i}A^{a_i}B^{b_i}C^{c_i}$, where $a_1,b_i,c_i=0,1$, is the elements of a Boolean ring and is very straightforward to manipulate the algebraic way.