well defined functions

[logistic_guy]

Determine whether the following functions \(\displaystyle f\) are well defined:

\(\displaystyle \bold{(a)} \ f \ : \ \mathbb{Q} \rightarrow \mathbb{Z}\) defined by \(\displaystyle f\left(\frac{a}{b}\right) = a\).

\(\displaystyle \bold{(b)}\ f \ : \ \mathbb{Q} \rightarrow \mathbb{Q}\) defined by \(\displaystyle f\left(\frac{a}{b}\right) = \frac{a^2}{b^2}\).

 

[khansaheb]

Determine whether the following functions \(\displaystyle f\) are well defined:

\(\displaystyle \bold{(a)} \ f \ : \ \mathbb{Q} \rightarrow \mathbb{Z}\) defined by \(\displaystyle f\left(\frac{a}{b}\right) = a\).

\(\displaystyle \bold{(b)}\ f \ : \ \mathbb{Q} \rightarrow \mathbb{Q}\) defined by \(\displaystyle f\left(\frac{a}{b}\right) = \frac{a^2}{b^2}\).

show us your effort/s to solve this problem.

 

[logistic_guy]

\(\displaystyle \bold{(a)}\)
Let us pick any rational number, say \(\displaystyle \frac{5}{6}\), and apply the definition.

\(\displaystyle f\left(\frac{5}{6}\right) = 5\)

We also know that \(\displaystyle \frac{5}{6} = \frac{5}{6} \times \frac{2}{2} = \frac{10}{12}\), so:

\(\displaystyle f\left(\frac{10}{12}\right) = 10\)

Since \(\displaystyle f\left(\frac{5}{6}\right) \neq f\left(\frac{10}{12}\right)\), then \(\displaystyle f\) is not well defined.