Logic & Simple Proof Methods: Lecture Notes

We can combine multiple propositions, which we'll conveniently denote as $p, q, r, s, t$, etc., using logical operators:

Operator	Name	Usage Example	$\LaTeX$	Read as...	True when...
$\neg$	Negation ('Not')	$\neg p$	$\neg p$	"Not $p$"	$p$ is false.
$\wedge$	Conjunction ('And')	$p \wedge q$	$p \wedge q$	"$p$ and $q$"	both $p$ and $q$ are true.
$\vee$	Disjunction ('Or')	$p \vee q$	$p \vee q$	"$p$ or $q$"	$p$ or $q$ or both are true.
$\oplus$	Exclusive Or ('Xor')	$p \oplus q$	$p \oplus q$	"$p$ xor $q$"	$p$ or $q$ are true, but not both.
$\rightarrow$	Implication	$p \rightarrow q$	$p \rightarrow q$	"$p$ implies $q$"	$p$ is false or $q$ is true.
$\leftrightarrow$	biconditional	$p \leftrightarrow q$	$p \leftrightarrow q$	"$p$ if and only if $q$"	$p$ and $q$ are both true or both false.

In some books, implication is denoted as $p \Rightarrow q$ (

$p \Rightarrow q$

) and the biconditional as $p \Leftrightarrow q$ (

$p \Leftrightarrow q$

), but, in our course, we'll use $\Rightarrow$ and $\Leftrightarrow$ later for a different purpose that also related to logic.