##Let  $p$ and $q$ be propositions, i.e. statements that have a
definite truth value, either $T$ (true) or $F$ (false).

#$p \wedge q$ is true

$\Leftrightarrow$ both $p$ and $q$ are true

#$p \vee q$ is true

$\Leftrightarrow$ at least one of $p$ or $q$ is true

#$\neg p$ is true

$\Leftrightarrow$ $p$ is false

#$p \to q$ is true

$\Leftrightarrow$ it is not the case that $p$ is true and $q$ is false

#$p \leftrightarrow q$ is true

$\Leftrightarrow$ $p$ and $q$ have the same truth value

#$p \oplus q$ is true

$\Leftrightarrow$ exactly one of $p$ or $q$ is true

##Let $P(x)$ be a predicate, i.e. some statement that becomes a
proposition when the variable $x$ is assigned a value (from some
implicit universe $U$ of discourse)

#$\forall xP(x)$ is true

$\Leftrightarrow$ $P(a)$ is true for all $a\in U$

#$\exists xP(x)$ is true

$\Leftrightarrow$ $P(a)$ is true for at least one $a\in U$

#$\neg \forall  x P(x)$

$\Leftrightarrow$ $\exists x \neg P(x)$

#$\neg \exists  x P(x)$

$\Leftrightarrow$ $\forall x \neg P(x)$