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An Introduction to Basic Set Theory

Composite Functions

Explore the concept of composite functions defined between sets, including their element-wise construction and examples. Understand the importance of function order, why some compositions are undefined, and the associative property. Gain practical experience by computing composite functions using Python, ensuring you grasp both theory and application.

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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 ...