We have inverse functions for addition. This is subtraction.
For function \(\oplus\), its inverse is \(\oplus'\), as defined below:
\(a\oplus b=c\)
\(b=c\oplus 'a\)
\(f(a,b)=c\rightarrow f^{-1}(c,b)=a\)
\(a+b=c\rightarrow b=c-a\)
There is no natural number \(b\) that satisfies:
\(3+b=2\)
While addition and multiplication are defined across all natural numbers, subtraction is not.
Subtraction is not commutative:
\(x-y\ne y-x\)
Subtraction is not associative:
\(x-(y-z)\ne (x-y)-z\)
We have inverse functions for multiplication. This is division.
These will not necessarily have solutions for natural numbers or integers.
\(a.b=c\rightarrow b=\dfrac{c}{a}\)
Division is not commutative:
\(\dfrac{x}{y}\ne \dfrac{y}{x}\)
\(\dfrac{x}{\dfrac{y}{z}}\ne \dfrac{\dfrac{x}{y}}{z}\)
Division is not left distributive over subtraction:
\(\dfrac{a}{b-c} \ne \dfrac{a}{b} -\dfrac{a}{c}\)
Division is right distributive over subtraction:
\(\dfrac{a-b}{c} =\dfrac{a}{b} -\dfrac{b}{c}\)