In the category , a function is defined as a set of ordered pairs that satisfies the following:
- If and
- If ,
- Then .
In other words, is itself a set! Specifically, it is a subset of the Cartesian Product of and where we presumably were taking our original elements. (, and .) So, .
- The domain of is the set of all objects , such that for some object .
- The range is the set of all objects such that for some object .
If is the domain of , and is a set that (at least) contains the range of (that is, ), then we say that is a function from into . We can write this as: .
Now go lift something heavy,