**Proposition:** Let and be sets and . Let .

Then the following hold:

- If , then .
- .
- .
- .
- for , .
- For all , implies that is
*onto*. - For all , implies that is
*one-to-one*.

*Note: is the “pre-image” of and is the inverse of .*

*Now go lift something heavy,*

Nick Horton