Operators

Basic Mathematical Operators

Some operators used in basic arithmetic and algebra.

Note that the × for ∖[Cross] is distinguished by being drawn slightly smaller than the × for ∖[Times].

Interpretation of some operators in basic arithmetic and algebra.

Operators in Calculus and Algebra

Operators used in calculus.

Operators for complex numbers and matrices.

Logical and Other Connectives

Operators used as logical connectives.

The operators ∧, ∨, and ¬ are interpreted as corresponding to the built-in functions (RefLink[And, paclet:ref/And]), (RefLink[Or, paclet:ref/Or]), and (RefLink[Not, paclet:ref/Not]), and are equivalent to the keyboard operators &&, ||, and !. The operators ⊻, ⊼, and ⊽ correspond to the built-in functions (RefLink[Xor, paclet:ref/Xor]), (RefLink[Nand, paclet:ref/Nand]), and (RefLink[Nor, paclet:ref/Nor]). Note that ¬ is a prefix operator.

x=>y and x⥰y  are both taken to give the built-in function (RefLink[Implies, paclet:ref/Implies])[x,y]. x∈y gives the built-in function (RefLink[Element, paclet:ref/Element])[x,y].

This is interpreted using the built-in functions (RefLink[And, paclet:ref/And]) and (RefLink[Implies, paclet:ref/Implies]):

3<4 ∧ x>5 \[Implies] y<7

The Wolfram Language supports most of the standard syntax used in mathematical logic. In the Wolfram Language, however, the variables that appear in the quantifiers ∀, ∃, and ∄ must appear as subscripts. If they appeared directly after the quantifier symbols then there could be a conflict with multiplication operations.

∀ and ∃ are essentially prefix operators like ∂:

(Subscript[∀, x])(Subscript[∃, y])ϕ[x,y]//FullForm

Operators Used to Represent Actions

Operators typically used to represent actions. All the operators except ∖[Square] are infix.

Following the Wolfram Language's usual convention, all the operators in the table are interpreted to give functions whose names are exactly the names of the characters that appear in the operators.

The operators are interpreted as functions with corresponding names:

x ⊕ y ⌢ z//FullForm

All the operators in the table above, except for □, are infix, so that they must appear in between their operands.

Bracketing Operators

Characters used as bracketing operators.

Interpretations of bracketing operators.

Operators Used to Represent Relations

Operators usually used to represent similarity or equivalence.

The special character == (or ∖[Equal]) is an alternative input form for ==. ≠ is used both for input and output:

{a==b,a == b,a!=b,a ≠ b}

Operators usually used for ordering by magnitude.

Operators used for relations in sets.

Operators usually used for other kinds of orderings.

Relational operators based on vertical bars.

Operators Based on Arrows and Vectors

Operators based on arrows are often used in pure mathematics and elsewhere to represent various kinds of transformations or changes.

-> is equivalent to ->:

x+y/.x -> 3

Arrow-like operators with built-in meanings in the Wolfram Language.

Ordinary arrows.

Vectors and related arrows.

All the arrow and vector-like operators in the Wolfram Language are infix:

x ⇌ y ⥮ z

Tees.