Def/Boolean conjunction

=AND= The truth value of "$S$ AND $T$" depends upon the truth value of $S$ and the truth value of $T$, as displayed in the following "truth table":

=OR=

The truth value of "$S$ OR $T$" depends upon the truth value of $S$ and the truth value of $T$, as displayed in the following "truth table":

Observe that the word "OR", and the symbol $\vee$, are used for an "inclusive or". In other words, if $S$ is true, and $T$ is true, then the sentence "$S$ OR $T$" is true. This contrasts with some common English usage, in which the word "or" is meant to express "one option or the other, but not both". In mathematics, and more commonly, in computer science, the boolean conjunction "XOR" is used for this "exclusive or".

=XOR= The truth value of "$S$ XOR $T$" depends upon the truth value of $S$ and the truth value of $T$, as displayed in the following "truth table":

=IMPLIES= The truth value of "$S$ IMPLIES $T$" (usually written $S \Rightarrow T$) depends upon the truth value of $S$ and the truth value of $T$, as displayed in the following "truth table":

Is a sentence of implication, of the form $S \Rightarrow T$, the sentence $S$ is called the defines::hypothesis or defines::premise, and $T$ the defines::conclusion of the implication. Therefore, an implication is a true sentence, if its premise is false (see truth by false premise ) or if its premise and conclusion are both true.