Share
Explore BrainMass

Proving one-to-one and onto properties

College level proof before real analysis. Please explain each step of your solution. Thank you.

Attachments

Solution Preview

Please see the attached file.

VI.
((b), (c), and (d) only.)

Solution.

(b) We can choose a codomain B={a, b, c, d, e} and define a function from A to B as follows.

f(1)=a; f(2)=b, f(3)=c and f(4)=d.

Then we can see clearly that f(x)=f(y) implies that x=y. So, f is one-to-one. But f is NOT onto, as for y=e, there is no element x in A such that f(x)=y=e.

(c) We can choose a codomain B={a, b, c, d} and define a function from A to B as follows.

f(1)=a; f(2)=b, f(3)=c and f(4)=d.

It is easy to see that f is one-to-one; and onto.

(d) We can choose a codomain B={a, b, c, d} and define a function from A to B as follows.

f(1)=a; f(2)=a, f(3)=c and f(4)=d.

Note that f is NOT one-to-one, as f(1)=f(2)=a. Moreover, f is NOT onto, as for y=b, there is no element x in A such that f(x)=y=b.

VII.

((b) and (d) only.)

Solution.

(b) Consider A={1,2,3}, ...

Solution Summary

This solution contains a detailed explanation of showing that a function is one-to-one and onto.

$2.19