Question

Verify associativity for the following three mappings : f : N → Z₀ (the set of non-zero integers), g : Z₀ → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = e^x.

Answer 1

Given that f : N → Z₀ , g : Z₀ → Q and h : Q → R .
gof : N → Q and hog : Z₀ → R
$\Rightarrow$h o (gof ) : N → R and (hog) o f: N → R
So, both have the same domains.
$$\begin{gathered} \left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(2x\right)=\frac{1}{2x}...\left(1\right) \\ \left(hog\right)\left(x\right)=h\left(g\left(x\right)\right)=h\left(\frac{1}{x}\right)=e^{\frac{1}{x}}...\left(2\right) \\ \text{Now,} \\ \left(ho\left(gof\right)\right)\left(x\right)=h\left(\left(gof\right)\left(x\right)\right)=h\left(\frac{1}{2x}\right)=e^{\frac{1}{2x}}[\text{from}\left(1\right)\text{]} \\ \left(\left(hog\right)of\right)\left(x\right)=\left(hog\right)\left(f\left(x\right)\right)=\left(hog\right)\left(2x\right)=e^{\frac{1}{2x}}[\text{from}\left(2\right)\text{]} \\ \text{⇒}\left(ho\left(gof\right)\right)\left(x\right)=\left(\left(hog\right)of\right)\left(x\right),\forall x\in N \\ \text{So,}ho\left(gof\right)=\left(hog\right)of \end{gathered}$$
Hence, the associative property has been verified.