1. Logics for Data and Knowledge Representation Introduction to Algebra Chiara Ghidini, Luciano Serafini, Fausto Giunchiglia and Vincenzo Maltese

2. Roadmap  Set theory  Basic notions  Operations  Properties  Relations  Functions 2

3. Describing the world 3 KimbaSimba Cita Hunts Eats Monkey Lion Near individuals sets relations

4. 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. 4 1 3 5 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

5. 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 5 1 3 5 7 9 A = {1, 3, 5, 7, 9} A = { x | x is an odd number < 10} A SETS :: RELATIONS :: FUNCTIONS

6. 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 6 SETS :: RELATIONS :: FUNCTIONS

7. 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| = nthen |P(A)| = 2 n 7 SETS :: RELATIONS :: FUNCTIONS

8. 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 8 AB a b c d AB a b c d A  B A  B SETS :: RELATIONS :: FUNCTIONS

9. 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. 9 AB a b d A - B c A _A_A U SETS :: RELATIONS :: FUNCTIONS

10. 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 ? 10 SETS :: RELATIONS :: FUNCTIONS

11. Properties of sets  A  A = AA  A = A  A   =  A   = A  A  B = B  AA  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

12. 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)} 12 SETS :: RELATIONS :: FUNCTIONS

13. Relations  A (binary) relation R from set A to set B is a subset of A x B R  A x BxRy 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} 13 b B a (a,b) ∈ R A SETS :: RELATIONS :: FUNCTIONS

14. Relations (cont.)  An n-ary relation R n is a subset of A 1 x … x A n 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} 14 b B a (b, a) ∈ R -1 A SETS :: RELATIONS :: FUNCTIONS

15. Properties of relations Let 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; 15 SETS :: RELATIONS :: FUNCTIONS

16. 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. 16 SETS :: RELATIONS :: FUNCTIONS

17. 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 17 SETS :: RELATIONS :: FUNCTIONS

18. 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. 18 SETS :: RELATIONS :: FUNCTIONS

19. 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

20. 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} 20 SETS :: RELATIONS :: FUNCTIONS