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Logics for Data and Knowledge Representation. Introduction to Algebra. Chiara Ghidini, Luciano Serafini, Fausto Giunchiglia and Vincenzo Maltese. Roadmap. Set theory Basic notions Operations Properties Relations Functions. Describing the world. individuals. Cita. Monkey. sets. - PowerPoint PPT Presentation
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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
2
Describing the world
3
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.
19
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