An Introduction to Basic Set Theory/

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Composite Functions

Composite Functions

Learn about composite functions.

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Let’s assume we have two functions, f:A→Bf : A\to B and g:B→Cg: B\to C, defined for some arbitrary sets A,B,A,B, and CC. Here is how the composite function from AA to CC, written as g∘f:A→Cg\circ f : A \to C, will be defined:

It’s important to note that for every x∈Ax\in A, we have its image y∈Cy\in C under the function g∘fg\circ f.

Examples

Let’s consider the following sets:

A={a,b,c,d,e}B={α,β,γ}C={j,k,l,m,n}D={u,v,w,x,y,z}\begin{align*} A &= \{a, b, c, d, e\}\\ B&=\{\alpha , \beta ,\gamma \}\\ C&=\{j, k, l, m, n\}\\ D&=\{u, v, w, x, y, z\}\end{align*}

Let’s also consider the functions f:A→B, g:B→C,f: A\to B, \: g: B\to C, and h:C→Dh:C\to D, which are defined as follows:

f={(a,α),(b,β),(c,γ),(d,α),(e,β)}g={(α,j),(β,m),(γ,n)}h={(j,z),(k,y),(l,x),(m ...