Exclusive Disjunction#
Definition: Exclusive Disjunction
The exclusive disjunction function is the Boolean operator \(\mathop{\operatorname{XOR}}: \{0, 1\}^2 \to \{0,1\}\) defined in the following way:
\[ \mathop{\operatorname{XOR}}(x, y) \overset{\text{def}}{=} \begin{cases} 1 \qquad \text{if } x \ne y \\ 0 \qquad \text{otherwise}\end{cases} \]
Notation
It is much more common to write \(\mathop{\operatorname{XOR}}(x, y)\) in the following way:
\[ x \oplus y \]
Theorem: Associativity of Exclusive Disjunction
The exclusive disjunction is associative:
\[ (x \oplus y) \oplus z = x \oplus (y \oplus x) \]
Proof
TODO
Theorem: Commutativity of Exclusive Disjunction
The exclusive disjunction is commutative:
\[ x \oplus y = y \oplus x \]
Proof
TODO