Mathematical Logic Symbols

logical operationsquantifiersset operationsset theory cardinalityrelation symbolsspecial symbolstemporal modal logicadvanced logical operationsalgebraic logicprobability logicproof notationsmiscellaneoustype theory lambda calculus

symbol	latex code	explanation
¬	\neg	Logical NOT
∧	\land	Logical AND
∨	\lor	Logical OR
⊕	\oplus	Logical XOR
→	\to	Implication (if...then...)
↔	\leftrightarrow	Biconditional (logical equivalence)
⊨	\models	Entails (satisfiability or semantic entailment)
⊢	\vdash	Provable (syntactic entailment)
⊤	\top	Tautology (always true)
⊥	\bot	Contradiction (always false)
∀	\forall	Universal quantifier (for all)
∃	\exists	Existential quantifier (there exists)
∄	\nexists	Negated existential quantifier (there does not exist)
∈	\in	Element of
∉	\notin	Not an element of
⊆	\subseteq	Subset or equal
⊂	\subset	Proper subset
⊇	\supseteq	Superset or equal
⊃	\supset	Proper superset
∩	\cap	Intersection (common elements)
∪	\cup	Union (all elements)
∖	\setminus	Set difference
⊆	\subseteq	Subset or equal (includes equality)
⊇	\supseteq	Superset or equal (includes equality)
⊄	\not\subset	Not subset
⊅	\not\supset	Not superset
ℵ₀	\aleph_0	Aleph-null (smallest infinite cardinal)
𝒫(A)	\mathcal{P}(A)	Power set (set of all subsets)
=	=	Equality
≠	\neq	Not equal
<	<	Less than
>	>	Greater than
≤	\leq	Less than or equal to
≥	\geq	Greater than or equal to
∅	\emptyset	Empty set (no elements)
ℕ	\mathbb{N}	Set of natural numbers
ℤ	\mathbb{Z}	Set of integers
ℚ	\mathbb{Q}	Set of rational numbers
ℝ	\mathbb{R}	Set of real numbers
ℂ	\mathbb{C}	Set of complex numbers
□	\Box	Necessity (it is necessary that)
◇	\Diamond	Possibility (it is possible that)
◊	\lozenge	Eventually (temporal logic)
○	\bigcirc	Next (temporal logic)
⊢	\vdash	Provable in proof system
⊬	\nvdash	Not provable in proof system
⊨	\models	Semantic entailment
⊭	\nvDash	Not semantic entailment
⊣	\dashv	Assertion of incompatibility
∧	\wedge	Meet (lattice intersection)
∨	\vee	Join (lattice union)
⊥	\bot	Bottom (least element)
⊤	\top	Top (greatest element)
Pr(A)	\Pr(A)	Probability of event A
E[X]	\mathbb{E}[X]	Expected value of random variable X
⇒	\Rightarrow	Implies in proofs
⇐	\Leftarrow	Implied by in proofs
○	\circ	Composition of functions
∞	\infty	Infinity
∴	\therefore	Therefore (consequence)
∵	\because	Because (reasoning)
λ	\lambda	Function abstraction in lambda calculus
⊢α	\vdash_\alpha	Type derivability in a type system
∼	\sim	Beta equivalence in lambda calculus
≡	\equiv	Alpha equivalence in lambda calculus
Γ	\Gamma	Context or typing environment in type systems