Operations in mathematics

An operation is a mapping that associates one or more elements of a set (argument) with another element (value). The term “operation” is usually applied to arithmetic or logical actions, in contrast to the term “operator”, which is more often applied to some mappings of a set onto itself that have remarkable properties.

Definition

Operation - mapping whose domain is the direct product of several sets. Mathematically, the operation can be written as ( and may coincide), where called the arity of the operation.

Related Definitions

Main article: Arnost

Operations differ in the number of sets whose Cartesian product is its domain of definition. For example, an operation can be unary if it maps one element of a set to one element of a set, or binary if it matches one element of a set to two elements of a set.

An algebraic operation is an operation. , in which and where - arity, i.e. . [one]

Properties

Operations may or may not have different properties. For example:

• Commutativity (switching property) - property of the operation " "When .
• Anticommutativity - for example, the subtraction operation, because .
• Associativity (combination property) - property of the operation " "When .
• Distributivity (distributive property) - for example, the operation of addition with respect to multiplication, since .
• Transitivity - operation " "Transitive if of ratios and follows that .
• Idempotency - if a repeated operation no longer changes the object, for example, taking modulo, for .
• Additivity - if for function right that , the function is additive.
• Multiplicativity - if for a function right that , the function is multiplicative.

Operations

Arithmetic

• Addition - binary operation, for example .
• Subtraction is the inverse operation of addition, for example .
• Multiplication is a hyperoperation of addition, for example .
• The division is the inverse of the multiplication operation, for example .
• Exponentiation - hyperoperation multiplication, for example .
• Root extraction - reverse operation to the degree, for example .
• Logarithm is the second inverse of exponentiation operation, for example .
• Tetration - exponentiation hyperoperation, for example .
• Superroot and superlog are reverse tetration operations.

Addition and negation are elementary arithmetic operations. All other, more complex operations are obtained as a result of hyperoperations. Thus, addition and subtraction are attributed to first-stage operations; multiplication and division - to the operations of the second stage; exponentiation, root extraction and logarithm — to third stage operations; Tetration and its inverse operations are rarely used operations of the fourth step, but this hyper-operation can be continued indefinitely, up to operations of the 5th, 6th and higher stages.

Mathematical analysis

• Differentiation — finding the derivative of a function, for example .
• Integration - back to differentiation; finding primitive function for example .
• Sampiration — finding a function by its roots.

brain teaser

Logical operations are operations on elements from a set of two elements: “true” and “false”, or “1” and “0”.

• Denial ( ) - unary operation; converts "1" to "0", and "0" to "1".
• Conjunction ( ) - binary operation; returns “1” only if both arguments are “1”.
• Disjunction ( ) - binary operation; returns "0", only if both arguments are "0".

Notes

1. ↑ Mathematical encyclopedia. - M .: Soviet encyclopedia. I.M. Vinogradov. 1977-1985.

see also

• Operations research
• Hyperoperator