## Logic Notation

implication

$P \implies Q$

P \implies Q
implication 2

$P \Rightarrow Q$

P \Rightarrow Q

The right arrow is sometimes used to indicate "implication".

implication 3

$P \to Q$

P \to Q

The single line arrow may be used to indicate "implication".

implication reverse

$P \Leftarrow Q$

P \Leftarrow  Q
implication reverse 2

$P \gets Q$

P \gets  Q

A single arrow to the left.

equivalence

$P \iff Q$

P \iff Q

\iff means "if and only if".

equivalence 2

$P \Leftrightarrow Q$

P \Leftrightarrow Q

The left right double arrow may be used to indicate "equivalence".

equivalence 3

$P \leftrightarrow Q$

P \leftrightarrow Q

The left right arrow may be used to indicate "equivalence".

equivalence 4

$P \equiv Q$

P \equiv Q

\equiv comes from "equivalence".

therefore

$\therefore$

\therefore
because

$\because$

\because
for all

$\forall x$

\forall x

This is Turned A.

for some

$\exists x$

\exists x

This is Turned E.

not exists

$\nexists$

\nexists

\nexists come from "not" and "exists".

quantifier sample

$\begin{eqnarray} & & {}^\forall \varepsilon \gt 0, {}^\exists \delta \gt 0 \mbox{ s.t. } \\ & & {}^\forall x \in \mathbb{ R }, 0 \lt |x - a| \lt \delta \implies |f(x) - b| \lt \varepsilon \end{eqnarray}$

\begin{eqnarray}
& & {}^\forall \varepsilon \gt 0, {}^\exists \delta \gt 0 \mbox{ s.t. } \\
& & {}^\forall x \in \mathbb{ R }, 0 \lt |x - a| \lt \delta
\implies |f(x) - b| \lt \varepsilon
\end{eqnarray}

This is the (ε, δ)-definition of limit.

logical conjunction

$P \land Q$

P \land Q

\land comes from "And in logic".

logical disjunction

$P \lor Q$

P \lor Q

\lor comes from "Or in logic".

negation

$\lnot P$

\lnot P

\lnot comes from "Not in logic".

negation 2

$\overline{ P }$

\overline{ P }

Another way to show negation is to draw a line over the letter.

negation 3

$!P$

!P

You can also write an "!" in front of the letter to indicate negation.

exclusive disjunction

$P \oplus Q$

P \oplus Q

A + ("plus") sign in a circle ("O") indicates an exclusive disjunction.

exclusive disjunction 2

$P \veebar Q$

P \veebar Q

This symbol is a combination of the letters V ("vee") and horizontal line ("bar"), and is sometimes used to represent an exclusive disjunction.

exclusive disjunction sample

$P \oplus Q = (P \land \lnot Q) \lor (\lnot P \land Q)$

P \oplus Q = (P \land \lnot Q) \lor (\lnot P \land Q)

This is a formula relating exclusive disjunction to logical disjunction, logical product, and negation.

tautology

$\top$

\top

This is used to show tautology. There's a horizontal line on top.

$\bot$

\bot

This is used to show contradiction. There's a horizontal line on bottom.

provable

$P \vdash Q$

P \vdash Q

\vdash comes from "vertical line" and "dash".

logical consequence

$P \models Q$

P \models Q

The same notation applies when using \vDash instead of \models.

Formula
Symbol
Text
Other