Math 2.0

General algebraic systems Definition

Associativity

Builds on Binary operation

Definition

Let SS be a set and let ∗\ast be a binary operation on SS. The operation ∗\ast is associative when (a∗b)∗c=a∗(b∗c)(a \ast b) \ast c = a \ast (b \ast c) for all a,b,c∈Sa, b, c \in S.

Remarks

Addition and multiplication of real numbers are associative. Subtraction of integers is not associative: (1−1)−1=−1(1 - 1) - 1 = -1, but 1−(1−1)=11 - (1 - 1) = 1. Composition of maps from a set to itself is associative. For an associative operation, each way to put parentheses in a1∗a2∗⋯∗ana_1 \ast a_2 \ast \cdots \ast a_n gives the same result.

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Binary operation, definition, no evidence levelBinary operationdefinitionAssociativity, definition, no evidence levelAssociativitydefinition
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  1. Binary operation definition
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12 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Associative_property, Definition
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