Discrete Mathematics FunctionsDefinition: A function from a set to a set , denoted : that assigns each element to exactly one ele → is a well-defined ruleif is the unique element of assigned by the f he element of . Exampl : Recall: Let and be sets. The Cartesian product of and all ordered p

Know how to prove \ (f\) is an onto function. To show that a function is not onto, all we need is to find an element \ (y\in B\), and show that no \ (x\)-value from \ (A\) would satisfy \ (f (x)=y\). In addition to finding images & preimages of elements, we also find images & preimages of sets.

Proving a Function is Onto. Let y be a general element of . Y You should not be using any information about the function at this point. Find x ∈ X such that . f (x) = y Finding x may involve scratchwork. Find a counterexample. Example 7.2.7. Arrow Diagram: Not Onto. Figure 7.2.8. A function which is not onto Y

Onto functions focus on the codomain. We want to know if it contains elements not associated with any element in the domain. A function \ (f : {A}\to {B}\) is onto if, for every element \ (b\in B\), there exists an element \ (a\in A\) such that \ [f (a) = b.\]