# Thread: Mappings question - Urgent

1. ## Mappings question - Urgent

Hi,

The question is:

Let S, T and U be sets and let a and b be mappings from S to T and from T to U respectively. Assume that the mapping a is not onto.
a) Give an example of sets S, T and U and mappings a and b, such that b o a is onto.
b)Can the mapping b o a be invertible, and, if yes, which additional properties do a and b possess?
c)If the answer to (b) is yes, give examples of sets S, T and U and of mappings a and b for which b o a is invertible and for which it is not. (the answer to (a) may be one of the examples. The sets s, T and U need not be the same as in (a))

Can anybody help with one please?

a) A basic example: $\displaystyle S=\{1\}, T=\{1,2\}, U=\{1\};\ a:1\mapsto 1$ (there is no choice for $\displaystyle b$, what is it?)
b)In the given example, $\displaystyle b\circ a$ is invertible. Why?
Furthermore, $\displaystyle b\circ a$ is invertible iff it is onto and one-to-one.
We've just seen that $\displaystyle a$ is not forced to be onto. What about $\displaystyle b$? And has $\displaystyle a$ to be injective?