Images and preimages
Given a function , and — a subset of domain, then the set of all possible values given by on arguments from , we call the image under as is denoted as . So:
Given a subset of the set of values , the sets of all arguments such that their values are in , is called the preimage of under and we denote it as . So:
Let , . Then is the set of all even numbers. If is the set of even numbers, then is the set of all natural numbers divisible by . If is the set of odd numbers, then . On the other hand, is the set of even numbers.
Let now , . Then . On the other hand, , and .
Let now , . Then . And .
Let finally , . Therefore, , so it is the line . On the other hand, .