Definition 0.4.1.
Let be non-empty sets. A function from to is a rule or formula that takes elements of as inputs and returns elements of as outputs. We write this as
and if takes as an input and returns then we write this as Every function must satisfy the following two conditions
- The function must be defined on every possible input from the set
That is, no matter which element we choose, the function must return an element so that - The function is only allowed to return one result for each input. So if we find that
1
You may have learned this in the context of plotting functions on the Cartesian plane, as “the vertical line test”. If the graph intersects a vertical line twice, then the same -value will give two -values and so the graph does not represent a function. and then the only way that can be a function is if is exactly the same as