Theory of Computer Science A2. Mathematical Foundations Gabriele R¨oger

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Theory of Computer Science

A2. Mathematical Foundations

Gabriele R¨ oger

University of Basel

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Theory of Computer Science

— A2. Mathematical Foundations

A2.1 Sets, Tuples, Relations A2.2 Functions

A2.3 Summary

Gabriele R¨oger (University of Basel) Theory of Computer Science 2 / 17

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A2.1 Sets, Tuples, Relations

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Sets

I set: unordered collection of distinguishable objects;

each object contained at most once I notations:

I explicit, listing all elements, e. g. A = {1, 2, 3}

I implicit, specifying a property characterizing all elements, e. g. A = {x | x ∈ N and 1 ≤ x ≤ 3}

I implicit, as a sequence with dots, e. g. Z = {. . . , −2, −1, 0, 1, 2, . . . }

I e ∈ M: e is in set M (an element of the set) I e ∈ / M: e is not in set M

I empty set ∅ = {}

I cardinality |M | of a finite set M : number of elements in M German: Menge, Element, leere Menge, M¨ achtigkeit/Kardinalit¨ at

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Sets

I A ⊆ B: A is a subset of B,

i. e., every element of A is an element of B I A ⊂ B: A is a strict subset of B ,

i. e., A ⊆ B and A 6= B.

I power set P (M ): set of all subsets of M e. g., P({a, b}) =

I Cardinality of power set of finite set S : |P(S )| =

German: Teilmenge, echte Teilmenge, Potenzmenge

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Set Operations

I intersection A ∩ B = {x | x ∈ A and x ∈ B}

A B

I union A ∪ B = {x | x ∈ A or x ∈ B }

A B

I difference A \ B = {x | x ∈ A and x ∈ / B}

A B

I complement A = B \ A, where A ⊆ B and

B is the set of all considered objects (in a given context) A

German: Schnitt, Vereinigung, Differenz, Komplement

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Tuples

I k -tuple: ordered sequence of k objects I written (o 1 , . . . , o _k ) or ho ₁ , . . . , o _k i I unlike sets, order matters (h1, 2i 6= h2, 1i) I objects may occur multiple times in a tuple I objects contained in tuples are called components I terminology:

I k = 2: (ordered) pair I k = 3: triple

I more rarely: quadruple, quintuple, sextuple, septuple, . . . I if k is clear from context (or does not matter),

often just called tuple

German: k -Tupel, Komponente, Paar, Tripel

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Cartesian Product

I for sets M ₁ , M ₂ , . . . , M _n , the Cartesian product M 1 × · · · × M n is the set

M ₁ × · · · × M _n = {ho ₁ , . . . , o _n i | o ₁ ∈ M ₁ , . . . , o _n ∈ M _n }.

I Example: M 1 = {a, b, c}, M 2 = {1, 2},

M ₁ × M ₂ = {ha, 1i, ha, 2i, hb, 1i, hb, 2i, hc , 1i, hc , 2i}

I special case: M ^k = M × · · · × M (k times) I example with M = {1, 2}:

M ² = {h1, 1i, h1, 2i, h2, 1i, h2, 2i}

German: kartesisches Produkt

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Relations

I an n-ary relation R over the sets M 1 , . . . , M n

is a subset of their Cartesian product: R ⊆ M ₁ × · · · × M _n . I example with M = {1, 2}:

R ≤ ⊆ M ² as R ≤ = {h1, 1i, h1, 2i, h2, 2i}

German: (n-stellige) Relation

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Exercise

Consider S = P({1, 2}) × {a, b}.

1

Write down three different elements of S.

2

What is |S |?

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A2.2 Functions

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Functions

Definition (Total Function)

A (total) function f : D → C (with sets D, C ) maps every value of its domain D

to exactly one value of its codomain C .

German: (totale) Funktion, Definitionsbereich, Wertebereich Example

I square : Z → Z with square(x) = x ² I add : N ² ₀ → N 0 with add(x, y) = x + y I add _R : R ² → R with add _R (x, y) = x + y

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Functions: Example

Example

Let Q = {q ₀ , q 1 , q 2 , q accept , q reject } and Γ = {0, 1, }.

Define δ : (Q \ {q _accept , q _reject }) × Γ → Q × Γ × {L, R} by

δ 0 1

q ₀ hq ₀ , 0, Ri hq ₀ , 1, Ri hq ₁ , , Li

q ₁ hq ₂ , 1, Li hq ₁ , 0, Li hq _reject , 1, Li

q 2 hq ₂ , 0, Li hq ₂ , 1, Li hq _accept , , Ri

Then, e. g., δ(q ₀ , 1) = hq ₀ , 1, Ri

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Partial Functions

Definition (Partial Function)

A partial function f : X → _p Y maps every value in X to at most one value in Y .

If f does not map x ∈ X to any value in Y , then f is undefined for x.

German: partielle Funktion Example

f : N 0 × N 0 → _p N 0 with

f (x, y) =

( x − y if y ≤ x undefined otherwise

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Exercises

Let V = {X , Y , Z }, Σ = {a, b, c } and Q = {q ₁ , q 2 } be three sets.

1

Specify a non-trivial example for a partial function δ : Q × Σ → _p P(Q).

2

Specify a non-trivial example for a

relation P ⊆ (V ∪ Σ) ² × V ² .

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A2.3 Summary

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Summary

I sets: unordered, contain every element at most once I tuples: ordered, can contain the same object multiple times I Cartesian product: M ₁ × · · · × M _n set of all n-tuples

where the i-th component is in M i

I function f : X → Y maps every value in X to exactly one value in Y

I partial function g : X → _p Y may be undefined

for some values in X

Theory of Computer Science A2. Mathematical Foundations Gabriele R¨oger

Theory of Computer Science

A2. Mathematical Foundations

Gabriele R¨ oger

University of Basel

Theory of Computer Science

— A2. Mathematical Foundations

A2.1 Sets, Tuples, Relations A2.2 Functions

A2.3 Summary

A2.1 Sets, Tuples, Relations

Sets

I set: unordered collection of distinguishable objects;

each object contained at most once I notations:

I explicit, listing all elements, e. g. A = {1, 2, 3}

I implicit, specifying a property characterizing all elements, e. g. A = {x | x ∈ N and 1 ≤ x ≤ 3}

I implicit, as a sequence with dots, e. g. Z = {. . . , −2, −1, 0, 1, 2, . . . }

I e ∈ M: e is in set M (an element of the set) I e ∈ / M: e is not in set M

I empty set ∅ = {}

I cardinality |M | of a finite set M : number of elements in M German: Menge, Element, leere Menge, M¨ achtigkeit/Kardinalit¨ at

Sets

I A ⊆ B: A is a subset of B,

i. e., every element of A is an element of B I A ⊂ B: A is a strict subset of B ,

i. e., A ⊆ B and A 6= B.

I power set P (M ): set of all subsets of M e. g., P({a, b}) =

I Cardinality of power set of finite set S : |P(S )| =

German: Teilmenge, echte Teilmenge, Potenzmenge

Set Operations

I intersection A ∩ B = {x | x ∈ A and x ∈ B}

A B

I union A ∪ B = {x | x ∈ A or x ∈ B }

A B

I difference A \ B = {x | x ∈ A and x ∈ / B}

A B

I complement A = B \ A, where A ⊆ B and

B is the set of all considered objects (in a given context) A

German: Schnitt, Vereinigung, Differenz, Komplement

Tuples

I k -tuple: ordered sequence of k objects I written (o 1 , . . . , o k ) or ho 1 , . . . , o k i I unlike sets, order matters (h1, 2i 6= h2, 1i) I objects may occur multiple times in a tuple I objects contained in tuples are called components I terminology:

I k = 2: (ordered) pair I k = 3: triple

I more rarely: quadruple, quintuple, sextuple, septuple, . . . I if k is clear from context (or does not matter),

often just called tuple

German: k -Tupel, Komponente, Paar, Tripel

Cartesian Product

I for sets M 1 , M 2 , . . . , M n , the Cartesian product M 1 × · · · × M n is the set

M 1 × · · · × M n = {ho 1 , . . . , o n i | o 1 ∈ M 1 , . . . , o n ∈ M n }.

I Example: M 1 = {a, b, c}, M 2 = {1, 2},

M 1 × M 2 = {ha, 1i, ha, 2i, hb, 1i, hb, 2i, hc , 1i, hc , 2i}

I special case: M k = M × · · · × M (k times) I example with M = {1, 2}:

M 2 = {h1, 1i, h1, 2i, h2, 1i, h2, 2i}

German: kartesisches Produkt

Relations

I an n-ary relation R over the sets M 1 , . . . , M n

is a subset of their Cartesian product: R ⊆ M 1 × · · · × M n . I example with M = {1, 2}:

R ≤ ⊆ M 2 as R ≤ = {h1, 1i, h1, 2i, h2, 2i}

German: (n-stellige) Relation

Exercise

Consider S = P({1, 2}) × {a, b}.

Write down three different elements of S.

What is |S |?

A2.2 Functions

Functions

Definition (Total Function)

A (total) function f : D → C (with sets D, C ) maps every value of its domain D

to exactly one value of its codomain C .

German: (totale) Funktion, Definitionsbereich, Wertebereich Example

I square : Z → Z with square(x) = x 2 I add : N 2 0 → N 0 with add(x, y) = x + y I add R : R 2 → R with add R (x, y) = x + y

Functions: Example

Example

Let Q = {q 0 , q 1 , q 2 , q accept , q reject } and Γ = {0, 1, }.

Define δ : (Q \ {q accept , q reject }) × Γ → Q × Γ × {L, R} by

δ 0 1

q 0 hq 0 , 0, Ri hq 0 , 1, Ri hq 1 , , Li

q 1 hq 2 , 1, Li hq 1 , 0, Li hq reject , 1, Li

q 2 hq 2 , 0, Li hq 2 , 1, Li hq accept , , Ri

Then, e. g., δ(q 0 , 1) = hq 0 , 1, Ri

Partial Functions

Definition (Partial Function)

A partial function f : X → p Y maps every value in X to at most one value in Y .

If f does not map x ∈ X to any value in Y , then f is undefined for x.

German: partielle Funktion Example

I k -tuple: ordered sequence of k objects I written (o 1 , . . . , o _k ) or ho ₁ , . . . , o _k i I unlike sets, order matters (h1, 2i 6= h2, 1i) I objects may occur multiple times in a tuple I objects contained in tuples are called components I terminology:

I for sets M ₁ , M ₂ , . . . , M _n , the Cartesian product M 1 × · · · × M n is the set

M ₁ × · · · × M _n = {ho ₁ , . . . , o _n i | o ₁ ∈ M ₁ , . . . , o _n ∈ M _n }.

M ₁ × M ₂ = {ha, 1i, ha, 2i, hb, 1i, hb, 2i, hc , 1i, hc , 2i}

I special case: M ^k = M × · · · × M (k times) I example with M = {1, 2}:

M ² = {h1, 1i, h1, 2i, h2, 1i, h2, 2i}

is a subset of their Cartesian product: R ⊆ M ₁ × · · · × M _n . I example with M = {1, 2}:

R ≤ ⊆ M ² as R ≤ = {h1, 1i, h1, 2i, h2, 2i}

I square : Z → Z with square(x) = x ² I add : N ² ₀ → N 0 with add(x, y) = x + y I add _R : R ² → R with add _R (x, y) = x + y

Let Q = {q ₀ , q 1 , q 2 , q accept , q reject } and Γ = {0, 1, }.

Define δ : (Q \ {q _accept , q _reject }) × Γ → Q × Γ × {L, R} by

q ₀ hq ₀ , 0, Ri hq ₀ , 1, Ri hq ₁ , , Li

q ₁ hq ₂ , 1, Li hq ₁ , 0, Li hq _reject , 1, Li

q 2 hq ₂ , 0, Li hq ₂ , 1, Li hq _accept , , Ri

Then, e. g., δ(q ₀ , 1) = hq ₀ , 1, Ri

A partial function f : X → _p Y maps every value in X to at most one value in Y .

f : N 0 × N 0 → _p N 0 with

Let V = {X , Y , Z }, Σ = {a, b, c } and Q = {q ₁ , q 2 } be three sets.

Specify a non-trivial example for a partial function δ : Q × Σ → _p P(Q).

relation P ⊆ (V ∪ Σ) ² × V ² .

I sets: unordered, contain every element at most once I tuples: ordered, can contain the same object multiple times I Cartesian product: M ₁ × · · · × M _n set of all n-tuples

I partial function g : X → _p Y may be undefined