Class 9: Cardinality of Finite Sets

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

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

Transcript

1. Class 9: Set Cardinality cs2102: Discrete Mathematics | F16 uvacs2102.github.io David Evans | University of Virginia
2. Plan Relation Practice, Inversions Cardinality of Finite Sets Powersets Well-Ordering Practice Chicken-Egg Challenge: I have received a couple good contenders, but will keep the challenge open (without posting or selecting “winners”) until next Monday.
3. : , , ⊆ × PS3: ⟶
4. function: ≤ 1 out injective: ≤ 1 in total: ≥ 1 out surjective: ≥ 1 in PS2 Grades: ⟶
5. Composing Relations PS2 Submissions: ⟶ PS2 Grades: ⟶
6. Composing Relations ; : , , ; ⊆ × < ∘ ; ∷= < : , , < ⊆ ×
7. Composing Relations ; : , , ; ⊆ × < ∘ ; ∷= , , ;< ⊆ × , ∈ ;< ⟺ , ∈ ; ∧ , ∈ < < : , , < ⊆ ×
8. Composing Relations PS2 Submissions: ⟶ PS2 Grades: ⟶ PS2 Scores: ⟶ ::=
9. 8 7 6.5 7.5 5.5 13 10 9 13 16 21 30 61 19 5 6
10. Inverting a Relation The inverse of a relation R is defined by reversing all the arrows: E;: ⟶ , E; ⊆ × The inverse of : ⟶ , ⊆ × is:
11. Inverting a Relation The inverse of a relation R is defined by reversing all the arrows: E;: ⟶ , E; ⊆ × , ∈ E; ⟺ , ∈ The inverse of : ⟶ , ⊆ × is:
12. Inverse of < <: ℤ ⟶ ℤ, I ⊆ ℤ×ℤ , ∈ E; ⟺ , ∈
13. Inverse of = =: ℤ ⟶ ℤ, J ⊆ ℤ×ℤ , ∈ E; ⟺ , ∈
14. Slack break: any questions so far , ∈ E; ⟺ , ∈ Are there any relations : ℤ ⟶ ℤ other than =, where E; = ?
15. Other Self-Inverses? , ∈ E; ⟺ , ∈
17. Cardinality of Finite Sets
18. Set Cardinality If is a finite set, the cardinality of , written ||, is the number of elements in . Is this a totally satisfying definition?
19. Set Cardinality If is a finite set, the cardinality of , written ||, is the number of elements in . Fundamental set operation: membership ∈ Can we define cardinality in terms of membership?
20. Alternate Definition The cardinality of the set ℕM = ∈ ℕ ∧ < } is .
21. Alternate Definition The cardinality of the set M = ∈ ℕ ∧ < } is . If there is a bijection between two sets, they have the same cardinality.
22. Example What is the cardinality of { 1, 2, 4, 8 } ? The cardinality of the set M = ∈ ℕ ∧ < } is . If there is a bijection between two sets, they have the same cardinality.
23. Relations and Sizes function: ≤ 1 out injective: ≤ 1 in total: ≥ 1 out surjective: ≥ 1 in If there is a surjective relation between and what do we know about their sizes?
24. Relations and Sizes function: ≤ 1 out injective: ≤ 1 in total: ≥ 1 out surjective: ≥ 1 in If there is a surjective function between and what do we know about their sizes?
25. Relations and Sizes function: ≤ 1 out injective: ≤ 1 in total: ≥ 1 out surjective: ≥ 1 in If there is a total surjective injective function between and what do we know about their sizes?
26. Union Cardinality What is the cardinality of ∪ ? The cardinality of the set M = ∈ ℕ ∧ < } is . If there is a bijection between two sets, they have the same cardinality.
27. Power Set The power set of A is the set of all subsets of A. ∈ ⟺ ⊆
28. Power Set The power set of A is the set of all subsets of A. ∈ ⟺ ⊆ Is ∅ ∈ ?
29. Power Set The power set of A is the set of all subsets of A. ∈ ⟺ ⊆ Is ∈ ?
30. Size of Power Set = 2|Z|
31. Well-Ordering Proofs
32. Well-Ordering Proofs ∷= ℕ\ = 2\ 1. ∷= ∈ ℕ ℕ\ ≠ 2\ } 2. Assume is non-empty. 3. By well-ordering principle, there must be some smallest element, ∈ .
33. 1. ∷= ∈ ℕ ℕ\ ≠ 2\ } 2. Assume is non-empty. 3. By well-ordering principle, there must be some smallest element, ∈ .
34. 1. ∷= ∈ ℕ ℕ\ ≠ 2\ } 2. Assume is non-empty. 3. By well-ordering principle, there must be some smallest element, ∈ . I’ll finish this proof on Thursday, but see if you can do it yourself.
35. Charge • PS4 Due Friday (6:29pm) • Thursday: Start Induction (MCS 5.1)