1. MAT 2720 Discrete Mathematics
Section 3.1
Functions

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

3. 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

4. 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

5. Notations

6. From Continuous to Discrete
Arrow Diagram Terminology:

7. Is this a Function? (I)

8. Is this a Function? (II)

9. One-to-One Functions

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

11. Equivalent Criteria

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

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

14. 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.

15. Onto Functions

16. Equivalent Criteria

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

18. 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

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

20. 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

21. Counting Problems…

22. Counting Problems…

23. Bijection

24. Inverse Functions

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