Logic Notation

implication

$P \implies Q$

P \implies Q

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implication 2

$P \Rightarrow Q$

P \Rightarrow Q

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

implication 3

$P \to Q$

P \to Q

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

implication reverse

$P \Leftarrow Q$

P \Leftarrow  Q

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implication reverse 2

$P \gets Q$

P \gets  Q

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A single arrow to the left.

equivalence

$P \iff Q$

P \iff Q

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\iff means "if and only if".

equivalence 2

$P \Leftrightarrow Q$

P \Leftrightarrow Q

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

equivalence 3

$P \leftrightarrow Q$

P \leftrightarrow Q

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

equivalence 4

$P \equiv Q$

P \equiv Q

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\equiv comes from "equivalence".

therefore

$\therefore$

\therefore

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because

$\because$

\because

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for all

$\forall x$

\forall x

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This is Turned A.

for some

$\exists x$

\exists x

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This is Turned E.

not exists

$\nexists$

\nexists

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\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}

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

logical conjunction

$P \land Q$

P \land Q

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\land comes from "And in logic".

logical disjunction

$P \lor Q$

P \lor Q

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\lor comes from "Or in logic".

negation

$\lnot P$

\lnot P

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\lnot comes from "Not in logic".

negation 2

$\overline{ P }$

\overline{ P }

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

negation 3

$!P$

!P

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

exclusive disjunction

$P \oplus Q$

P \oplus Q

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

exclusive disjunction 2

$P \veebar Q$

P \veebar Q

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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)

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

tautology

$\top$

\top

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

contradiction

$\bot$

\bot

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

provable

$P \vdash Q$

P \vdash Q

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

logical consequence

$P \models Q$

P \models Q

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