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Associativity

Associativity - раздел Математика, A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right Main Article: Associative Property The Associative Property I...

Main article: Associative property

The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms doesn't change. In contrast, the commutative property states that the order of the terms does not affect the final result.

Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function

which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example, but ).

Symmetry in mathematics

Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as a binary function then the resulting function is symmetric across the line y = x. As an example, if we let a function f represent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function, which can be seen in the image on the right.

For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then .

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A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right

A is a proper subset of B and conversely B is a proper superset of A... In set theory and related branches of mathematics a collection F of subsets... If every member of set A is also a member of set B then A is said to be a subset of B written A B A is...

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Tuples as nested ordered pairs
Another way of formalizing tuples is as nested ordered pairs: 1. The 0-tuple (i.e. the empty tuple) is represented by the empty set

Non-injective and surjective
· ·

Definable relations
An n-ary relation R on the universe M of a structure is said to be defin

Definable relations
An n-ary relation R on the universe M of a structure is said to be defin

Characteristic properties of tuples
The general rule for the identity of two -tuples is

Examples Commutative operations in mathematics
Two well-known examples of commutative binary operations:[7] · The addition of real numbers is commutative, since

Equivalence class
Main article: Equivalence class The set of all a and b for which a ~ b holds make up an equivalence class of X by ~. Let

Definitions
A linear order on a set S is a (binary) relation < with the following properties: irreflexivity: x≮x; asymmetry: x<

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