Class 8: Relations and Functions

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class 8: Functions and Relations cs2102: Discrete Mathematics | F16 uvacs2102.github.io David Evans | University of Virginia
2. Plan Functions Paradox / PS3 Questions Relations
3. Function Mathematical Data Type Domain Codomain
4. Function Mathematical Data Type Domain 4 2102 { 1, 2, 3, …} : $ Codomain ARCH 6.042 : ⟶
5. Operation Set Function ∈ Membership Test
6. Operation Set Function ∈ Membership Test = () Association
7. Defining a Function
8. Defining a Function Domain (Set) Codomain (Set) Associate elements of Domain with elements of Codomain
9. Example: NOT Domain Codomain P NOT(P) T F F T
10. Example: FTFF Domain Codomain A B FTFF(A, B) T T F T F T F T F F F F
11. Cartesian Product Definition. A Cartesian product of sets 7 ×$ × ⋯ ×: is a set consisting of all possible sequences of elements where the th element is chosen from = . × = , ∈ , ∈ } 7 ×$ × ⋯ ×: = 7 , $,…,DE = ∈ = }
12. Cartesian Product Definition. A Cartesian product of sets 7 ×$ × ⋯ ×: is a set consisting of all possible sequences of elements where the th element is chosen from = . × = , ∈ , ∈ } 7 ×$ × ⋯ ×: = 7 , $,…,DE = ∈ = } How many elements are in ×?
13. Example: FTFF Domain Codomain A B FTFF(A, B) T T F T F T F T F F F F
14. Example: Doubling ∷= 2 Domain Codomain
15. Example: Absolute Value ∷= || Domain Codomain
16. Example: Division , ∷= / Domain Codomain
17. Example: Square Root ∷= Domain Codomain
18. Total and Partial A function is total is every element in the domain has an associated codomain element. A partial function may have domain elements with no associated codomain element.
19. Total? , ∷= /
20. Total? , ∷= / Domain ⟶ Codomain: ℤ × ℤ ⟶ ℝ
21. Total? , ∷= / Domain ⟶ Codomain: ℤ × ℤU ⟶ ℝ
23. Functions in Programming Java Python java.util.Arrays.sort Math.sqrt System.out.println (int n) -> n + 1 math.sqrt list.sort print lambda n: n + 1
24. Mathematical Functions Programming Functions
25. Mathematical Functions cannot do this!
26. Slack break… Questions about PS3 topics Questions about functions What is the smallest natural number that cannot be described in eleven words?
27. What is the smallest natural number that cannot be described in eleven words?
28. The smallest natural number that cannot be described in eleven words. 1 2 3 4 5 6 7 8 9 10 11 What is the smallest natural number that cannot be described in eleven words?
31. Relations
32. Binary Relation
33. Binary Relation Is a binary relation a function?
34. Relation Properties A B
35. Relation Properties A B function: ≤ 1 out total: ≥ 1 out injective: ≤ 1 in surjective: ≥ 1 in
36. Relation Properties A B function: ≤ 1 out total: ≥ 1 out injective: ≤ 1 in surjective: ≥ 1 in bijective: = 1 out, = 1 in
37. : , , ⊆ × =
38. : , , ⊆ × <
39. Charge • PS3 Due Friday (6:29pm) • Next week: – Cardinality (MCS 4.5) – Start Induction (MCS 5.1)