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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...
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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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