MAT 2720 Discrete Mathematics Section 3.1 Functions http://myhome.spu.edu/lauw

Goals 3.1 Functions – Applications in Counting 3.3 Relations – Extension of Functions 3.4 Equivalence Relation – A special type of relation

Goals Review and Renew the concept of functions How to show that a function is an One-to-one function (Injection) How to show that a function is an Onto function (Surjection) Typically, you have learned these in a pre-calculus class. And it was used in calculus II for the derivatives of the inverse functions

You Know a Lot About Functions You are supposed to know a lot… Domain, Range, Codomain Inverse Functions One-to-one, Onto Functions Composite Functions

Notations

From Continuous to Discrete Arrow Diagram Terminology:

Is this a Function? (I)

Is this a Function? (II)

One-to-One Functions

One-to-One Functions This is NOT an easy criteria to demo/prove a function is injective.

Equivalent Criteria

Example 1 Determine if the given function is 1-1. Prove your answer. Proof: Analysis

Example 2 Determine if the given function is 1-1. Prove your answer. Proof: Analysis

Example 2 Determine if the given function is 1-1. Prove your answer. KEY: Must spell out the precise reasons. Since ?≠??, but f(?)=f(??), f is not injective.

Onto Functions

Equivalent Criteria

Example 3 Determine if the given function is onto. Prove your answer. Proof: Analysis

Example 3 Determine if the given function is onto. Prove your answer. Proof: An counter example is difficult to explain in this case Use contradiction instead – which may have the same “feel” of a counter example Analysis

Example 4 Determine if the given function is onto. Prove your answer. Proof: Analysis

Template Analysis: Write down some cases to determine if a function is injective/surjective. To show injective/surjective, give a (direct) proof. To show NOT injective/surjective, use An counter example if it is easy to explain Use contradiction Other type of valid arguments

Counting Problems…

Counting Problems…

Bijection

Inverse Functions

Group Explorations Very fun to do. Keep the fun between you and your partner.