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Operations in mathematics

Lecture



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 Operations in mathematics - mapping whose domain is the direct product of several sets. Mathematically, the operation can be written as Operations in mathematics ( Operations in mathematics and Operations in mathematics may coincide), where Operations in mathematics 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. Operations in mathematics , in which Operations in mathematics and Operations in mathematics where Operations in mathematics - arity, i.e. Operations in mathematics . [one]

Properties

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

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

Operations

Arithmetic

  • Addition - binary operation, for example Operations in mathematics .
  • Subtraction is the inverse operation of addition, for example Operations in mathematics .
  • Multiplication is a hyperoperation of addition, for example Operations in mathematics .
  • The division is the inverse of the multiplication operation, for example Operations in mathematics .
  • Exponentiation - hyperoperation multiplication, for example Operations in mathematics .
  • Root extraction - reverse operation to the degree, for example Operations in mathematics .
  • Logarithm is the second inverse of exponentiation operation, for example Operations in mathematics .
  • Tetration - exponentiation hyperoperation, for example Operations in mathematics .
  • 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 Operations in mathematics .
  • Integration - back to differentiation; finding primitive function for example Operations in mathematics .
  • 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 ( Operations in mathematics ) - unary operation; converts "1" to "0", and "0" to "1".
  • Conjunction ( Operations in mathematics ) - binary operation; returns “1” only if both arguments are “1”.
  • Disjunction ( Operations in mathematics ) - 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

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