Class 9: Relations Set Cardinality cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans | University of Virginia

Plan Binary Relations Injective, Surjective, Function, Total Composition Inversions Cardinality of Finite Sets Powersets, Well-Ordering Practice PS4 due Friday PS5 due Sept 29 Exam 1: in-class on Oct 5 covers through Class 11

Binary Relation What makes a binary relation a function? A B

Relation Properties A B function: ≤ 1 out total: ≥ 1 out injective: ≤ 1 in surjective: ≥ 1 in

Relation Properties A B function: ≤ 1 out total: ≥ 1 out injective: ≤ 1 in surjective: ≥ 1 in bijective: = 1 out, = 1 in

Defining Function 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. function: ≤ 1 out total: ≥ 1 out Alternate definitions: “function”: =1 out “partial function”: <= 1 out

Total Function? , ∷= /

No content

: , , ⊆ × PS3: ⟶

Definitions Recap function: ≤ 1 out injective: ≤ 1 in total: ≥ 1 out surjective: ≥ 1 in Which of these properties should the PS3 relation satisfy? PS3: ⟶

function: ≤ 1 out injective: ≤ 1 in total: ≥ 1 out surjective: ≥ 1 in PS3 Grades: ⟶

Composing Relations PS3 Submissions: ⟶ PS3 Grades: ⟶

Composing Relations A : , , A ⊆ × B ∘ A ∷= B : __, __, B ⊆

Composing Relations A : , , A ⊆ × B ∘ A ∷= , , AB ⊆ × , ∈ AB ⟺ , ∈ A ∧ , ∈ B B : , , B ⊆ ×

Inverting a Relation The inverse of a relation R is defined by reversing all the arrows: JA: ⟶ , JA ⊆ × The inverse of : ⟶ , ⊆ × is:

Inverting a Relation The inverse of a relation R is defined by reversing all the arrows: JA: ⟶ , JA ⊆ × , ∈ JA ⟺ , ∈ The inverse of : ⟶ , ⊆ × is:

Inverse of < <: ℤ ⟶ ℤ, N ⊆ ℤ×ℤ , ∈ JA ⟺ , ∈

Inverse of = =: ℤ ⟶ ℤ, O ⊆ ℤ×ℤ , ∈ JA ⟺ , ∈

Slack break: any questions so far , ∈ JA ⟺ , ∈ Are there any relations : ℤ ⟶ ℤ other than =, where JA = ?

Other Self-Inverses? , ∈ JA ⟺ , ∈

No content

Cardinality of Finite Sets

Oridinal vs. Cardinal Numbers

Oridinal vs. Cardinal Numbers Ordinal Cardinal first one second two third three fourth four fifth five … … 29th 29 English

Oridinal vs. Cardinal Numbers Ordinal Cardinal first one second two third three fourth four fifth five … … 29th 29 Ordinal Cardinal erste eins zwei zweite drei dritte vier vierte fünf fünfte … … neunundzwanzig neunundzwanzigste English German Ordinal Cardinal முதல் ஒன்று இரண்டாம் இரண்டு மூன்றாம் மூன்று நான்காம் நான்கு ஐந்து … Tamil

Set Cardinality If is a finite set, the cardinality of , written ||, is the number of elements in . Is this a totally satisfying definition?

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?

Alternate Definition The cardinality of the set ℕR = ∈ ℕ ∧ < } is .

Charge PS4 Due Friday (6:29pm) Thursday: Start Induction (MCS 5.1)