Understanding Functions and Relations in Discrete Structures

1. (CSC 102) Discrete Structures Lecture 20

2. Previous Lecture Summery • Relations and Functions • Definition of Function • Examples of Functions

3. Functions

4. Today’s Lecture • One-to-One Function • Onto Function • Bijective Function (One-to-One correspondence) • Inverse Functions

5. Sum/difference of Functions Let F: R → R and G: R → R be functions. Define new functions F + G: R → R: For all x ∈ R, (F + G)(x) = F(x) + G(x) F and G must have same Domains and Codomains.

6. Equality of Functions Theorem: If F: X → Y and G: X → Y are functions, then F = G if, and only if, F(x) = G(x) for allx ∈ X. Example

8. One-to-One Functions

9. One-to-One Functions The Rule is ‘ADD 4’ 1 2 3 4 5 5 6 7 8 9 Codomain(R)={5, 6, 7, 8, 9,10} Dom (R) = {1, 2, 3, 4, 5}

10. One-to-One Functions Identifying One-to-One functions defined on sets

11. One-to-One Functions on Infinite Sets

12. One-to-One Functions on Infinite Sets

13. One-to-One Functions on Infinite Sets

14. One-to-One Functions on Infinite Sets

15. Onto Functions on Sets

16. Onto Functions on Sets

17. Onto Functions on Sets Identifying Onto Functions

18. Onto Functions on Infinite Sets

19. Onto Functions on Infinite Sets

20. Onto Functions on Infinite Sets To prove that f is onto, you must prove ∀y ∈ Y, ∃x ∈ X such that f (x) = y. • There exists real number x such that y = f(x)? • Does f really send x to y?

21. Onto Functions on Infinite Sets

22. Onto Functions on Infinite Sets

23. Onto Functions on Infinite Sets

24. One-to-One Correspondence (Bijection)

25. One-to-One Correspondence (Bijection)

26. Inverse Functions Theorem The function F-1 is called inverse function.

27. Finding an Inverse Function The function f : R → R defined by the formula f (x) = 4x − 1, for all real numbers x

28. Theorem

29. Lecture Summery • One-to-One Function • Onto Function • Bijective Function (One-to-One correspondence) • Inverse Functions