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# composition

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1 (relative product) A method of combining functions in a serial manner. The composition of two functions f : XY and g : YZ is the function h : XZ with the property that h(x) = g(f(x) This is usually written as g ° f. The process of performing composition is an operation between functions of suitable kinds. It is associative, and identity functions fulfill the role of units.

f : XY and g : YZ

h : XZ

h(x) = g(f(x)

If R denotes the set of real numbers and f : RR, f(x) = sin(x) g : RR, g(x) = x2 + 3 then f° g is the function h: h : RR, h(x) = sin(x2 + 3)

f : RR, f(x) = sin(x)

g : RR, g(x) = x2 + 3

h : RR, h(x) = sin(x2 + 3)

The idea of composition of functions can be extended to functions of several variables.

2 A subdivision of a positive integer n into parts a1, a2,… ak in which the ordering is significant and in whichn = a1 + a2 + … + akwhere each ai is a positive integer. It is thus similar to a partition (see covering) but in a partition the ordering is not significant. In general the number of compositions of n is 2n−1.

n = a1 + a2 + … + ak

3 A particular form of association between entities found in object-oriented approaches. The association is used to indicate a hierarchy of objects such that objects lower in the hierarchy are part of objects higher in the hierarchy. Thus the hierarchy indicates a component structure.

Subjects: Computing.

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