Logics for Data and KnowledgeRepresentation

Introduction to Algebra

Chiara Ghidini, Luciano Serafini, Fausto Giunchiglia and Vincenzo Maltese

Roadmap Set theory

Basic notions Operations Properties

Relations Functions

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Describing the world

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Kimba Simba

Cita

Hunts Eats

Monkey

LionNear

individualssetsrelations

Sets A set is a collection of elements The description of a set must be unambiguous and unique: it

must be possible to decide whether an element belongs to the set or not.

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1 35

7 9

The set of odd numbers < 10

The set of students in this

room

The set of lions in a certain zoo

SETS :: RELATIONS :: FUNCTIONS

Describing sets Listing: the set is described by

listing all its elements

Abstraction: the set is described through a common property of its elements

Venn Diagrams: graphical representation that supports the formal description

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1 35

7 9

A = {1, 3, 5, 7, 9}

A = { x | x is an odd number < 10}

A

SETS :: RELATIONS :: FUNCTIONS

Basic notions on sets Empty Set: the set with no elements;

A = { } A =

Membership: element a belongs to the set A;

A = {a, b, c} a A

Non membership: element a doesn't belong to the set A

A = {b, c} a A

Equality: the sets A and B contain the same elements;

A = {b, c}; B = {b, c} A = B

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SETS :: RELATIONS :: FUNCTIONS

Basic notions on sets (cont.) Inequality: the sets A and B contain the same elements;

A = {c}; B = {b, c} A ≠ B

Subset: all elements of A belong to B;

A = {c}; B = {b, c} A B

Proper subset: all elements of A belong to B and they are not the same

A B and A ≠ B then A B

Power set: the set of all the subsets of A A = {a, b} P(A) = {, {a}, {b}, {a, b}}

|A| = n then |P(A)| = 2n

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SETS :: RELATIONS :: FUNCTIONS

Operations on sets Union: the set containing the the

members of A or B

Intersection: the set containing the members of both A and B

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A B

a

bc d

A B

a

bc d

A B

A B

SETS :: RELATIONS :: FUNCTIONS

Operations on sets (cont.) Difference: the set containing the

members of A and not of B

Complement: given a universal set U, the complement of A is the set whose members are the members of U - A.

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A B

a

bd

A - Bc

A

_ A

U

SETS :: RELATIONS :: FUNCTIONS

Exercises Given A = {t, z} and B = {v, z, t}, say whether the following

statements are true or false: A B A B z A B v B {v} B v A - B

Given A = {a, b, c, d} and B = {c, d, f} Find a set X such that A B = B X. Is this set unique? Is there any set Y such that A Y = B ?

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SETS :: RELATIONS :: FUNCTIONS

Properties of sets A A = A A A = A A = A = A

A B = B A A B = B A (commutative)

(A B) C = A (B C)

(A B) C = A (B C) (associative)

A (B C) = (A B) (A C)

A (B C) = (A B) (A C) (distributive)

_____ _ _ A B = A B

_____ _ _

A B = A B (De Morgan laws)11

SETS :: RELATIONS :: FUNCTIONS

Cartesian product Cartesian product of A and B: the set of ordered couples (a, b)

where a is a member of A and b a member of B

A x B = {(a, b) : a A and b B}

Notice that A x B ≠ B x A

Example:

A = {a, b, c}, B = {s, t}

A x B = {(a, s), (a, t), (b, s), (b, t), (c, s), (c, t)}

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SETS :: RELATIONS :: FUNCTIONS

Relations A (binary) relation R from set A to set B is a subset of A x B

R A x B xRy indicates that (x, y) R

The domain of R is the set Dom(R) = {a A | b ∃ B s.t. aRb}

The co-domain of R is the set Cod(R) = {b B | a ∃ A s.t. aRb}

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bB

a

(a,b) ∈ R

A

SETS :: RELATIONS :: FUNCTIONS

Relations (cont.) An n-ary relation Rn is a subset of A1 x … x An

n is the arity of the relation

The inverse relation of R A x B is the relation R-1 B x A where:

R-1 = {(b, a) | (a, b) R}

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bB

a

(b, a) ∈ R-1

A

SETS :: RELATIONS :: FUNCTIONS

Properties of relationsLet R be a binary relation on A, i.e. R A x A. R is said to be:

reflexive iff aRa a ∀ A;symmetric iff aRb implies bRa a, b ∀ A;transitive iff aRb and bRc imply aRc a, b, c ∀ A;anti-symmetric iff aRb and bRa imply a = b a, b ∀ A;

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SETS :: RELATIONS :: FUNCTIONS

Equivalence relations Given R A x A, R is an equivalence relation iff it is reflexive,

symmetric and transitive.

A partition of a set A is a family F of non-empty subsets of A s.t.: the subsets are pairwise disjoint the union of all the subsets is the set A

Notice that each element of A belongs to exactly one subset in F.

Given ≡ equivalence relation on A and a A, the equivalence class of a is the set [a] = {x | a ≡ x}

Notice that if x [a] then [x] = [a]

The quotient set of A w.r.t. ≡ is the set {[x] | x A} which defines a partition of A.

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SETS :: RELATIONS :: FUNCTIONS

Order relations Given R A x A, R is a (partial) order relation iff it is reflexive,

anti-symmetric and transitive.

If the relation holds a, b ∀ A then it is a total order

If a, b ∀ A either aRb or bRa or a = b then it is a strict order

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SETS :: RELATIONS :: FUNCTIONS

Functions A function f from A to B is a binary relation that associates to

each element a in A exactly one element b in B.

f : A B

The image of an element a A is denoted with f(a) B

Notice that it can be the case that the same element in B is the image of several elements in A.

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SETS :: RELATIONS :: FUNCTIONS

Functions (cont.) f: A B is injective if for distinct elements in A there is a distinct

element in B:

∀ a, b A and a ≠ b then f(a) ≠ f(b)

f: A B is surjective if for each element in B there is at least one element in A:

∀ b B a ∃ A s.t. f(a) = b

f: A B is bijective if it is injective and surjective.

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SETS :: RELATIONS :: FUNCTIONS

Functions (cont.) If f: A B is bijective we can define its inverse function f-1: B A

Given two functions f: A B and g: B C, the composition of f and g is the function g ○ f : C such that:

g ○ f = {(a, g(f(a)) | a A}

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SETS :: RELATIONS :: FUNCTIONS