Let \(A,B,C\) be sets. The following lemma shows that the composition of relations preserves the properties of a function.
Let \(f:A\mapsto B\) and \(g:B\mapsto C\) be functions. The composition $g\circ f\subseteq A\times C$ of the corresponding relations $f\subseteq A\times B$ and $g\subseteq B\times C$ is also a function defined by
$$(g\circ f):A\to C,\;(g\circ f)(a):=g(f(a))$$
and called the composition of functions.
If, for instance, $A=\mathbb R\to \mathbb R, f(x)=x^2$ and $g:\mathbb R\to \mathbb R, g(x)=x+2$, then $$(g\circ f)(x)=g(f(x))=g(x^2)=x^2+2,$$ and $$(f\circ g)(x)=f(g(x))=f(x+2)=x^2+2{x}+4.$$
Proofs: 1
Definitions: 1 2 3 4 5 6 7
Examples: 8
Parts: 9
Proofs: 10 11 12 13 14 15 16 17 18 19
Propositions: 20 21 22 23 24 25 26 27 28 29 30