Relations

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Wax on … wax off … these are the basics

http://www.youtube.com/watch?v=3PycZtfns_U

Sets

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computerinformation information

computation

SetA set is a group of objects.

SetA set is a group of objects.

{10, 23, 32}

N = {0, 1, 2, … }

Z = {… , -2, -1, 0, 1, 2, … }

Ø

U

empty set

universe

SetA set is a group of objects.

{10, 23, 32}

N = {0, 1, 2, … }

Z = {… , -2, -1, 0, 1, 2, … }

10 ∈ {10, 23, 32}

-1 ∉ N

Ø

U

empty set

universe

Membershipa is a member of set A

Subset A⊆B

∀x:: x∈A ⇒ x∈B

Every member of A is also an element of B.

Subset A⊆B

∀x:: x∈A ⇒ x∈B

∅ ⊆ A.A ⊆ A.A = B ⇔ A ⊆ B ∧ B ⊆ A.

Every member of A is also an element of B.

Subset A⊆B

∀x:: x∈A ⇒ x∈B

∅ ⊆ A.A ⊆ A.A = B ⇔ A ⊆ B ∧ B ⊆ A.

Proper subset A⊂B

∀x:: A⊆B ∧ A≠B

A is a subset of B and not equal to B.

Every member of A is also an element of B.

Union A∪B

∀x:: x∈A ∨ x∈BA∪B={ x | x∈A or x∈B }

Union A∪B

∀x:: x∈A ∨ x∈BA∪B={ x | x∈A or x∈B }

Union A∪B

∀x:: x∈A ∨ x∈BA∪B={ x | x∈A or x∈B }

A ∪ B = B ∪ A.A ∪ (B ∪ C) = (A ∪ B) ∪ C.A ⊆ (A ∪ B).A ∪ A = A.A ∪ ∅ = A.A ⊆ B ⇔ A ∪ B = B.

Intersection A∩B

∀x:: x∈A ∧ x∈BA∩B={ x | x∈A and x∈B }

Intersection A∩B

∀x:: x∈A ∧ x∈BA∩B={ x | x∈A and x∈B }

Intersection A∩B

∀x:: x∈A ∧ x∈BA∩B={ x | x∈A and x∈B }

A ∩ B = B ∩ A.A ∩ (B ∩ C) = (A ∩ B) ∩ C.A ∩ B ⊆ A.A ∩ A = A.A ∩ ∅ = ∅.A ⊆ B ⇔ A ∩ B = A.

Complements A\B, A’

∀x:: x∈A ∧ x∉BA\B={ x | x∈A and x∉B }

Complements A\B, A’

∀x:: x∈A ∧ x∉BA\B={ x | x∈A and x∉B }

A \ B ≠ B \ A.A ∪ A′ = U.A ∩ A′ = ∅.(A′)′ = A.A \ A = ∅.U′ = ∅.∅′ = U.A \ B = A ∩ B′.

A ∩ U = AA ∪ ∅ = A

Neutral elements

A ∩ U = AA ∪ ∅ = A

Neutral elements

A ∩ ∅ = ∅

A ∪ U = U

Zero elements

A ∩ U = AA ∪ ∅ = A

Neutral elements

A ∩ ∅ = ∅

A ∪ U = U

Zero elements

A ∩ A = AA ∪ A = A

Idempotence

A ∩ U = AA ∪ ∅ = A

Neutral elements

A ∩ ∅ = ∅

A ∪ U = U

Zero elements

A ∩ A = AA ∪ A = A

Idempotence

A ∪ B = B ∪ AA ∩ B = B ∩ A

Commutativity

A ∩ U = AA ∪ ∅ = A

Neutral elements

A ∩ ∅ = ∅

A ∪ U = U

Zero elements

A ∩ A = AA ∪ A = A

Idempotence

A ∪ B = B ∪ AA ∩ B = B ∩ A

Commutativity

A ∩ (B ∩ C) = (A ∩ B) ∩ CA ∪ (B ∪ C) = (A ∪ B) ∪ C

Associativity

A ∩ U = AA ∪ ∅ = A

Neutral elements

A ∩ ∅ = ∅

A ∪ U = U

Zero elements

A ∩ A = AA ∪ A = A

Idempotence

A ∪ B = B ∪ AA ∩ B = B ∩ A

Commutativity

A ∩ (B ∩ C) = (A ∩ B) ∩ CA ∪ (B ∪ C) = (A ∪ B) ∪ C

Associativity

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Distributivity

A ∩ U = AA ∪ ∅ = A

Neutral elements

A ∩ ∅ = ∅

A ∪ U = U

Zero elements

A ∩ A = AA ∪ A = A

Idempotence

A ∩ A’ = ∅

A ∪ A’ = U

Complement

A ∪ B = B ∪ AA ∩ B = B ∩ A

Commutativity

A ∩ (B ∩ C) = (A ∩ B) ∩ CA ∪ (B ∪ C) = (A ∪ B) ∪ C

Associativity

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Distributivity

Similar to boolean algebra

a ∧ 1 = aa ∨ 0 = a

Neutral elements

a ∧ 0 = 0a ∨ 1 = 1

Zero elements

a ∧ a = aa ∨ a = a

Idempotence

a ∧ ¬ a = 0a ∨ ¬ a = 1

Negation

a ∨ b = b ∨ aa ∧ b = b ∧ a

Commutativity

a ∧ (b ∧ c) = (a ∧ b) ∧ ca ∨ (b ∨ c) = (a ∨ b) ∨ c

Associativity

a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

Distributivity

A ∩ U = AA ∪ ∅ = A

Neutral elements

A ∩ ∅ = ∅

A ∪ U = U

Zero elements

A ∩ A = AA ∪ A = A

Idempotence

A ∩ A’ = ∅

A ∪ A’ = U

Complement

A ∪ B = B ∪ AA ∩ B = B ∩ A

Commutativity

A ∩ (B ∩ C) = (A ∩ B) ∩ CA ∪ (B ∪ C) = (A ∪ B) ∪ C

Associativity

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Distributivity

A ∩ U = A A ∪ B = B ∪ AA ∪ ∅ = A

A ∩ ∅ = ∅

A ∪ U = U

A ∩ A = AA ∪ A = A

A ∩ A’ = ∅

A ∪ A’ = U

Neutral elements

Zero elements

Idempotence

Complement

A ∩ (B ∩ C) = (A ∩ B) ∩ C

A ∩ B = B ∩ A

A ∪ (B ∪ C) = (A ∪ B) ∪ C

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(A ∩ B)’ = (A’) ∪ (B’)(A ∪ B)’ = (A’) ∩ (B’)

Commutativity

Associativity

Distributivity

DeMorgan’s

A ⊆ A.

A ⊆ B ∧ B ⊆ A ⇔ A = B.

A ⊆ B ∧ B ⊆ C ⇔ A ⊆ C

Reflexivity

Anti-symmetry

Transitivity

Relations

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Scissors

Paper

Stone

Scissors

Paper

Stone

beats

beats

beats

Scissors

Paper

Stone

beats

beats

beats

beats Scissors Paper StoneScissors FALSE TRUE FALSEPaper FALSE FALSE TRUEStone TRUE FALSE FALSE

beats Scissors Paper StoneScissors FALSE TRUE FALSEPaper FALSE FALSE TRUEStone TRUE FALSE FALSE

beats = {(Scissors, Paper), (Paper, Stone), (Stone, Scissors)}

beats Scissors Paper StoneScissors FALSE TRUE FALSEPaper FALSE FALSE TRUEStone TRUE FALSE FALSE

beats = {(Scissors, Paper), (Paper, Stone), (Stone, Scissors)}

beats ⊆ {Scissor, Paper, Stone} x {Scissor, Paper, Stone}

Cartesian product AxB

AxB={ (a,b) | a∈A and b∈B }

Cartesian product AxB

AxB={ (a,b) | a∈A and b∈B }

A × ∅ = ∅.A × (B ∪ C) = (A × B) ∪ (A × C).(A ∪ B) × C = (A × C) ∪ (B × C).

N-ary Relation

A1, A2, ..., AnR ⊆ A1 x A2 x...x An

Binary Relation

A1, A2R ⊆ A1 x A2

(a,b) ∈ RaRb

Binary Relation

A1, A2R ⊆ A1 x A2

(a,b) ∈ RaRb

Binary Relation

A1, A2R ⊆ A1 x A2

(a,b) ∈ RaRb

dom R = {a⏐∃b :: (a,b) ∈ R}range R = {b⏐∃a :: (a,b) ∈ R}

Reflexive relationevery element x of A is in relation R with itself

∀x: x∈A: xRx

Reflexive relationevery element x of A is in relation R with itself

Symmetric relationif there is a relation between x and y, then there is a relation between y and x

∀x: x∈A: xRx

Reflexive relationevery element x of A is in relation R with itself

Symmetric relationif there is a relation between x and y, then there is a relation between y and x

∀x: x∈A: xRx

∀x,y: x,y∈A: xRy ⇒ yRx

Reflexive relationevery element x of A is in relation R with itself

Transitive relation...

Symmetric relationif there is a relation between x and y, then there is a relation between y and x

∀x: x∈A: xRx

∀x,y: x,y∈A: xRy ⇒ yRx

Reflexive relationevery element x of A is in relation R with itself

Transitive relation...

Symmetric relationif there is a relation between x and y, then there is a relation between y and x

∀x: x∈A: xRx

∀x,y: x,y∈A: xRy ⇒ yRx

∀x,y,z: x,y,z∈A: (xRy ∧ yRz) ⇒ xRz

Reflexive relationevery element x of A is in relation R with itself

Transitive relation...

Symmetric relationif there is a relation between x and y, then there is a relation between y and x

Eq

uiva

lent

rel

atio

n

∀x: x∈A: xRx

∀x,y: x,y∈A: xRy ⇒ yRx

∀x,y,z: x,y,z∈A: (xRy ∧ yRz) ⇒ xRz

Reflexive relationevery element x of A is in relation R with itself

Examples

=>, <≥, ≤beats

Examples

=>, <≥, ≤beats

reflexive, symmetric, transitive

transitive

reflexive, transitive

-

[x]R= {y | xRy}Equivalence class

[x]R= {y | xRy}Equivalence class

[1]= =

[x]R= {y | xRy}Equivalence class

[1]= = {1}

Example

Consider the relation ≡5 over the integer numbers Z defined as x≡5y if and only if x-y is a multiple of 5 (where x,y∈Z).Is ≡5 an equivalence relation?

Example

Consider the relation ≡5 over the integer numbers Z defined as x≡5y if and only if x-y is a multiple of 5 (where x,y∈Z).Is ≡5 an equivalence relation?

What is [1]≡5 ?

R ⊆ AxA{(a,b), (b,c), (c,d)}

a b c dR ⊆ AxA{(a,b), (b,c), (c,d)}

a b c dR ⊆ AxA{(a,b), (b,c), (c,d)}

a b c dR ⊆ AxA{(a,b), (b,c), (c,d)}

a b c dR ⊆ AxA{(a,b), (b,c), (c,d)}

a b c dR ⊆ AxA{(a,b), (b,c), (c,d)}

R1 = R;∀i:i>1:Ri = Ri-1 ∪ {(a,b) | ∃c:: (a,c)∈Ri-1 ∧ (c,b)∈Ri-1}.Rt = ∪i≥1Ri = R1 ∪ R2 ∪ R3 ∪ ...

Transitive closure

Irreflexive relationno element x of A is in relation R with itself

∀x: x∈A: ¬(xRx)

Irreflexive relationno element x of A is in relation R with itself

Antisymmetric relationif there is a relation between x and y and one between y and x, then x equals y

∀x: x∈A: ¬(xRx)

Irreflexive relationno element x of A is in relation R with itself

Antisymmetric relationif there is a relation between x and y and one between y and x, then x equals y

∀x: x∈A: ¬(xRx)

∀x,y: x,y∈A: (xRy ∧ yRx) ⇒ x=y

Irreflexive relationno element x of A is in relation R with itself

Asymmetric relationxRy and yRx cannot hold at the same time

Antisymmetric relationif there is a relation between x and y and one between y and x, then x equals y

∀x: x∈A: ¬(xRx)

∀x,y: x,y∈A: (xRy ∧ yRx) ⇒ x=y

Irreflexive relationno element x of A is in relation R with itself

Asymmetric relationxRy and yRx cannot hold at the same time

Antisymmetric relationif there is a relation between x and y and one between y and x, then x equals y

∀x: x∈A: ¬(xRx)

∀x,y: x,y∈A: (xRy ∧ yRx) ⇒ x=y

∀x,y: x,y∈A: xRy ⇒ ¬(yRx)

Irreflexive relationno element x of A is in relation R with itself

Examples

=>, <≥, ≤beats

Examples

=>, <≥, ≤beats

antisymmetric

irreflexive, asymmetric

antisymmetric

irreflexive

Non-symmetric relationa relation that is not symmetric

∀x,y: x,y∈A: (xRy) ∧ ¬(yRx)

Non-symmetric relationa relation that is not symmetric

∀x,y: x,y∈A: (xRy) ∧ ¬(yRx)

Non-symmetric relationa relation that is not symmetric

Total relationR is defined on the entire A.

∀x,y: x,y∈A: (xRy) ∧ ¬(yRx)

Non-symmetric relationa relation that is not symmetric

∀x,y: x,y∈A: xRy ∨ yRx

Total relationR is defined on the entire A.

Examples

=>, <≥, ≤beats

Examples

=>, <≥, ≤beats

-

non-symmetric

non-symmetric, total

-

Acyclic relationthere are no elements with transitive closure to themselves

Acyclic relationthere are no elements with transitive closure to themselves

∀n: n∈N:( ¬(∃x1, x2, ...,xn: x1, x2, ...,xn∈A: x1Rx2 ∧ x2Rx3 ∧ ... ∧ xn-1Rxn ∧ xnRx1 ) )

Acyclic relationthere are no elements with transitive closure to themselves

∀n: n∈N:( ¬(∃x1, x2, ...,xn: x1, x2, ...,xn∈A: x1Rx2 ∧ x2Rx3 ∧ ... ∧ xn-1Rxn ∧ xnRx1 ) )

>, <

Acyclic relationthere are no elements with transitive closure to themselves

∀n: n∈N:( ¬(∃x1, x2, ...,xn: x1, x2, ...,xn∈A: x1Rx2 ∧ x2Rx3 ∧ ... ∧ xn-1Rxn ∧ xnRx1 ) )

>, < acyclic

R is partial order;R is total relation.

Total order

R is reflexive;R is transitive;R is antisymmetric.

Partial order

R is reflexive;R is transitive.

Strict partial order

A, BF ⊆ A x B(a,b)∈F ∧ (a,c)∈F ⇒ b=cdomF = A

F:A -> B

Function

A, BF ⊆ A x B(a,b)∈F ∧ (a,c)∈F ⇒ b=cdomF = A

F:A -> B

Function

FºG(x) = F(G(x))Function composition

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