In order to reflect the more natural meaning of $x$ or $y$, which is "either $x$ or $y$, but not both", we introduce another Boolean function, called the exclusive disjunction "$\oplus$", (sometimes also called the "exclusive or").
For any given propositions $x$ and $y$, the exclusive disjunction $x\oplus y$ is defined using the conjunction, disjunction and negation:
$$x\oplus y:=(x\vee y)\wedge \neg(x\wedge y),$$
expressing
"$x$ or $y$, but not both."
The truth table of the exclusive disjunction is given by:
Truth Table of Exclusive Disjunction :------------- $[[x]]_I$| $[[y]]_I$| $[[x\oplus y]]_I$ $1$| $1$| $0$ $0$| $1$| $1$ $1$| $0$| $1$ $0$| $0$| $0$
