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Class 8: Functions and Relations cs2102: Discrete Mathematics | F16 uvacs2102.github.io David Evans | University of Virginia

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Plan Functions Paradox / PS3 Questions Relations

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Function Mathematical Data Type Domain Codomain

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Function Mathematical Data Type Domain 4 2102 { 1, 2, 3, …} : $ Codomain ARCH 6.042 : ⟶

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Operation Set Function ∈ Membership Test

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Operation Set Function ∈ Membership Test = () Association

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Defining a Function

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Defining a Function Domain (Set) Codomain (Set) Associate elements of Domain with elements of Codomain

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Example: NOT Domain Codomain P NOT(P) T F F T

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Example: FTFF Domain Codomain A B FTFF(A, B) T T F T F T F T F F F F

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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 = ∈ = }

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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 ×?

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Example: FTFF Domain Codomain A B FTFF(A, B) T T F T F T F T F F F F

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Example: Doubling ∷= 2 Domain Codomain

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Example: Absolute Value ∷= || Domain Codomain

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Example: Division , ∷= / Domain Codomain

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Example: Square Root ∷= Domain Codomain

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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.

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Total? , ∷= /

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Total? , ∷= / Domain ⟶ Codomain: ℤ × ℤ ⟶ ℝ

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Total? , ∷= / Domain ⟶ Codomain: ℤ × ℤU ⟶ ℝ

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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

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Mathematical Functions Programming Functions

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Mathematical Functions cannot do this!

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Slack break… Questions about PS3 topics Questions about functions What is the smallest natural number that cannot be described in eleven words?

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What is the smallest natural number that cannot be described in eleven words?

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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?

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Relations

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Binary Relation

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Binary Relation Is a binary relation a function?

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Relation Properties A B

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

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

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: , , ⊆ × =

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: , , ⊆ × <

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Charge • PS3 Due Friday (6:29pm) • Next week: – Cardinality (MCS 4.5) – Start Induction (MCS 5.1)