05 - relations
DESCRIPTION
I used this set of slides for the lecture on Relations I gave at the University of Zurich for the 1st year students following the course of Formale Grundlagen der Informatik.TRANSCRIPT
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Wax on … wax off … these are the basics
http://www.youtube.com/watch?v=3PycZtfns_U
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computerinformation information
computation
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SetA set is a group of objects.
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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
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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
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Subset A⊆B
∀x:: x∈A ⇒ x∈B
Every member of A is also an element of B.
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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.
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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.
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Union A∪B
∀x:: x∈A ∨ x∈BA∪B={ x | x∈A or x∈B }
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Union A∪B
∀x:: x∈A ∨ x∈BA∪B={ x | x∈A or x∈B }
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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.
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Intersection A∩B
∀x:: x∈A ∧ x∈BA∩B={ x | x∈A and x∈B }
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Intersection A∩B
∀x:: x∈A ∧ x∈BA∩B={ x | x∈A and x∈B }
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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.
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Complements A\B, A’
∀x:: x∈A ∧ x∉BA\B={ x | x∈A and x∉B }
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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′.
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A ∩ U = AA ∪ ∅ = A
Neutral elements
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A ∩ U = AA ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
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A ∩ U = AA ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = AA ∪ A = A
Idempotence
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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
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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
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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
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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
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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
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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
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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
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A ⊆ A.
A ⊆ B ∧ B ⊆ A ⇔ A = B.
A ⊆ B ∧ B ⊆ C ⇔ A ⊆ C
Reflexivity
Anti-symmetry
Transitivity
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Scissors
Paper
Stone
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Scissors
Paper
Stone
beats
beats
beats
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Scissors
Paper
Stone
beats
beats
beats
beats Scissors Paper StoneScissors FALSE TRUE FALSEPaper FALSE FALSE TRUEStone TRUE FALSE FALSE
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beats Scissors Paper StoneScissors FALSE TRUE FALSEPaper FALSE FALSE TRUEStone TRUE FALSE FALSE
beats = {(Scissors, Paper), (Paper, Stone), (Stone, Scissors)}
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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}
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Cartesian product AxB
AxB={ (a,b) | a∈A and b∈B }
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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).
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N-ary Relation
A1, A2, ..., AnR ⊆ A1 x A2 x...x An
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Binary Relation
A1, A2R ⊆ A1 x A2
(a,b) ∈ RaRb
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Binary Relation
A1, A2R ⊆ A1 x A2
(a,b) ∈ RaRb
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Binary Relation
A1, A2R ⊆ A1 x A2
(a,b) ∈ RaRb
dom R = {a⏐∃b :: (a,b) ∈ R}range R = {b⏐∃a :: (a,b) ∈ R}
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Reflexive relationevery element x of A is in relation R with itself
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∀x: x∈A: xRx
Reflexive relationevery element x of A is in relation R with itself
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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
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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
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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
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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
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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
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Examples
=>, <≥, ≤beats
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Examples
=>, <≥, ≤beats
reflexive, symmetric, transitive
transitive
reflexive, transitive
-
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[x]R= {y | xRy}Equivalence class
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[x]R= {y | xRy}Equivalence class
[1]= =
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[x]R= {y | xRy}Equivalence class
[1]= = {1}
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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?
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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 ?
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R ⊆ AxA{(a,b), (b,c), (c,d)}
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a b c dR ⊆ AxA{(a,b), (b,c), (c,d)}
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a b c dR ⊆ AxA{(a,b), (b,c), (c,d)}
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a b c dR ⊆ AxA{(a,b), (b,c), (c,d)}
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a b c dR ⊆ AxA{(a,b), (b,c), (c,d)}
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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
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Irreflexive relationno element x of A is in relation R with itself
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∀x: x∈A: ¬(xRx)
Irreflexive relationno element x of A is in relation R with itself
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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
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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
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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
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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
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Examples
=>, <≥, ≤beats
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Examples
=>, <≥, ≤beats
antisymmetric
irreflexive, asymmetric
antisymmetric
irreflexive
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Non-symmetric relationa relation that is not symmetric
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∀x,y: x,y∈A: (xRy) ∧ ¬(yRx)
Non-symmetric relationa relation that is not symmetric
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∀x,y: x,y∈A: (xRy) ∧ ¬(yRx)
Non-symmetric relationa relation that is not symmetric
Total relationR is defined on the entire A.
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∀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.
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Examples
=>, <≥, ≤beats
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Examples
=>, <≥, ≤beats
-
non-symmetric
non-symmetric, total
-
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Acyclic relationthere are no elements with transitive closure to themselves
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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 ) )
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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 ) )
>, <
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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
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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
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A, BF ⊆ A x B(a,b)∈F ∧ (a,c)∈F ⇒ b=cdomF = A
F:A -> B
Function
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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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Tudor Gîrbawww.tudorgirba.com
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