staymathematicsMedium,Associative law(associated laws) YesBinary operationA property that can exist means that in a representation containing more than two combinable operatorsoperatorIf the position of is not changed, the order of its operations will not affect the calculated value.[1]
Is calledCombinableIf it meets the followingAssociative law：[1]
。
The order of operations does not affect the value of the expression, and it can be proved that this can be used when there are "arbitrary" multiple
The expression of the operation is still valid.Therefore, when
When it is combinable, the order of operations does not need to be standardized without making its meaning unclear, so the parentheses can be omitted and simply written as:
However, it should be remembered that changing the order of operations does not include or allow changing the real operations by moving the operators in the expression.
give an example
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multiplication
Associative law of multiplication: Multiply three numbers. Multiply the first two numbers and then the third number, or multiply the last two numbers and then the first number. Their product remains unchanged.
Letters: (a × b) × c=a × (b × c)
Collective coalescence
Intersection of sets,Union operationBoth meet the combination law:[2]
complexandQuaternionThe addition and multiplication of can be combined.OctonionsThe addition of is also combinable, but the multiplication rule is not.
becauselinear transformation It is a function that can be expressed as a matrix, in which the function composition can be usedMatrix multiplicationIt is immediately known that the matrix multiplication is associative.
If M is a set and S is a set of all functions mapped from M to M, the operation of function composition on S is combinable:
。
More generally, given four sets M, N, P and Q, and
、
, then
Same as before.In short, the composition of maps is always combinable.
Given a set with three elements A, B and C, its operation is as follows:
A
B
C
A
A
A
A
B
A
B
C
C
A
A
A
It can be combined.However, this operation is not commutative.
Incorporatibility
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If a binary operation * on set S does not satisfy the associative law, it is calledIncoherent。The symbols are:[1]
。
Under this operation, the order of operation isyesAffected.subtraction、divisionandpowerAre simple examples of non combinable operations:
In general, when the non associative operation occurs more than once in a representation, parentheses must be used to represent itOperation order。however,mathematicianA special operation order rule will be adopted for several common non combinable operations.This is simply a syntax convention to reduce parentheses.
