Sets. Ordered pairs. Sets Sets Ordered PAIR The set theory is a branch of mathematical logic that was created by Georg Cantor in 1870s. Def. A set is a unordered collection of objects being regarded as a single object. Examples: A = 1, 2, 3, 4, 5 f g B = x A (x 3) (x is odd) f 2 j ≥ ^ g C = ?, A , B , A, B f f g f g f gg N = 0, 1, 2, 3, . f g p. 2 Sets Sets Ordered PAIR If x belongs to a set A, we say that it is a member (or an element) of A and write x A. 2 If x is not a member of A, we write x = A. 2 Empty set, denoted by ? or , has no members: ; x (x = ?). 8 2 p. 3 Sets Sets Ordered PAIR Two sets A and B are equal, A = B, iff they have exactly the same elements: x (x A x B) 8 2 $ 2 For any two objects x and y, we can make a set containing exactly these two objects x, y f g If those two objects are identical, x = y, we get a singleton set, x, x = x . f g f g Notice that these sets are not equal: ?, ? , ?, ? f g f f gg p. 4 Sets Sets Ordered PAIR Set-builder notation. A set of all objects that satisfy the property P: A = x P(x) f j g Examples: B = x Z x is even f 2 j g C = x Z k Z (x = 2k) f 2 j 9 2 g D = x (x Z) ( k Z (x = 2k)) f j 2 ^ 9 2 g E = 0, 2, 2, 4, 4, 6, 6, . f − − − g In naive set theory, any definable collection is a valid set. And usu- ally it works fine. However, it leads to contradictions, such as Russell’s paradox. p. 5 Russell’s PARADOX Sets Ordered PAIR Let R be the set of all sets that are not members of themselves: R = x x = x f j 2 g It is legal to ask, is R a member of itself or not. There are only two cases, either R R, or R = R. 2 2 So, where is the paradox? p. 6 Russell’s PARADOX Sets Ordered PAIR R = x x = x f j 2 g (a) If R is a member of itself, then by its definition, R = R, 2 R R R = R. 2 ! 2 (b) Otherwise, if R is not a member of itself, then R R, 2 R = R R R. 2 ! 2 Therefore, we get a contradiction, R R R = R. 2 $ 2 There exist several axiomatic systems that rule out such pathologi- cal cases. For example, Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). p. 7 Union AND Intersection Sets Ordered PAIR Union of two sets A and B, A B = x (x A) (x B) [ f j 2 _ 2 g 1, 2 2, 3 = 1, 2, 3 f g [ f g f g Intersection of two sets A and B, A B = x (x A) (x B) \ f j 2 ^ 2 g 1, 2 2, 3 = 2 f g \ f g f g p. 8 Difference Sets Ordered PAIR Difference between two sets A and B, A B = x (x A) (x = B) n f j 2 ^ 2 g A B 1, 2 2, 3 = 1 f g n f g f g p. 9 Complement Sets Ordered PAIR If there is a universal setU of all possible objects, then the comple- ment of a set A is A = U A = x U x = A = x U (x A) n f 2 j 2 g f 2 j : 2 g U A A Exmaple: When U = Z: Odd = x Z x is odd f 2 j g Even = Odd = Z Odd n p. 10 Set IDENTITIES Sets Ordered PAIR Given the universal set U, sets with respect to union, intersection, and complement satisfy the same identities as propositions with respect to , , and ^ _ : Sets: A BA B AU ? Propositions: p \ q p [ q pTF ^ _ : A = A A B = A B [ \ A B = A B \ [ A (B C) = (A B) C \ \ \ \ A (B C) = (A B) C [ [ [ [ A (B C) = (A B) (A C) \ [ \ [ \ A (B C) = (A B) (A C) [ \ [ \ [ See the full list in Rosen’s book. p. 11 Set IDENTITIES Sets Ordered PAIR Let’s prove De Morgan’s law for sets: A B = A B [ \ (x A B) = (x A B) 2 [ : 2 [ = ((x A) (x B)) : 2 _ 2 (x A B) = (x A) (x B) 2 \ 2 ^ 2 = (x A) (x B) : 2 ^ : 2 = ((x A) (x B)) : 2 _ 2 The propositions in the right hand sides of the equations are equal, therefore, the left hand sides are equal too. p. 12 Subset Sets Ordered PAIR A is a subset of B iff every element of A is an element of B: A B x((x A) (x B)) ⊆ $ 8 2 ! 2 1, 2 1, 2, 3 f g ⊆ f g 1, 2, 3 1, 2, 3 f g ⊆ f g ? 1, 2, 3 ⊆ f g A is a proper subset of B iff A is a subset of B, but it’s not equal to B A B x((x A) (x B)) x((x B) (x = A)) $ 8 2 ! 2 ^ 9 2 ^ 2 ( 1, 2 1, 2, 3 f g f g ? 1, 2, 3 (f g Proper subset A is strictly “smaller” than B. ( p. 13 POWER SET Sets Ordered PAIR Def. The set of all subsets of A is called a power set of A, denoted by (A). P Examples: ( 0, 1 ) = ?, 0 , 1 , 0, 1 P f g f g f g f g ( 0, 1, 2 ) = ?, 0 , 1 , 2 , 0, 1 , 0, 2 , 1, 2 , 0, 1, 2 P f g f g f g f g f g f g f g f g p. 14 POWER SET Sets Ordered PAIR Def. The set of all subsets of A is called a power set of A, denoted by (A). P Examples: ( 0, 1 ) = ?, 0 , 1 , 0, 1 P f g f g f g f g ( 0, 1, 2 ) = ?, 0 , 1 , 2 , 0, 1 , 0, 2 , 1, 2 , 0, 1, 2 P f g f g f g f g f g f g f g f g p. 15 Cardinality Sets Ordered PAIR Def. The cardinality of a finite set A is equal to the number of ele- ments in A. It’s denoted by A . j j ? = 0 j j 0, 1, 2, 3, 4 = 5 jf gj We already know the subtraction rule for the cardinality of a union: A B = A + B A B j [ j j j j j − j \ j p. 16 Question Sets Ordered PAIR Compute the cardinality of the power set (A) if A = n. P j j In other words, since the power set is the set of all subsets, the task is to count the number of subsets of a set with n elements. A n (A) = 2j j = 2 jP j p. 17 Question Sets Ordered PAIR Compute the cardinality of the power set (A) if A = n. P j j In other words, since the power set is the set of all subsets, the task is to count the number of subsets of a set with n elements. A n (A) = 2j j = 2 jP j p. 18 Question Sets Ordered PAIR Compute the cardinality of the power set (A) if A = n. P j j In other words, since the power set is the set of all subsets, the task is to count the number of subsets of a set with n elements. A n (A) = 2j j = 2 jP j p. 19 GenerALIZED UNION AND INTERSECTION Sets Ordered PAIR Union of a finite set of sets: [ A0 = A0 f g n [ [ A0, A1,... An = Ai = A0 A1 ... An f g i=0 [ [ [ Intersection of a finite set of sets: \ A0 = A0 f g n \ \ A0, A1,... An = Ai = A0 A1 ... An f g i=0 \ \ \ p. 20 Ordered PAIR Sets Ordered PAIR Def. The ordered pair of a A and b B is an ordered collection (a, b). 2 2 Two ordered pairs are equal (a, b) = (c, d) if and only if (a = c) (b = d). ^ Observe that this property implies that (a, b) = (b, a) 6 unless a = b. So, the order matters. This is why it is called the ordered pair, and (a, b) is not equivalent to a set a, b . f g (1, 2) = (1, 2) (1, 2) = (1, 3) 6 (1, 2) = (2, 1) 6 p. 21 Ordered PAIR Sets Ordered PAIR More examples of ordered pairs: (1, 2) ( 1 , 2 ) f g f g (1, 2, 3, 4, 5 ) f g ((1, 2), ?) If a A and b B, what is the set of all ordered pairs (a, b)? 2 2 p. 22 Cartesian PRODUCT Sets Ordered PAIR Def. Let A and B be sets. The Cartesian product of A and B, denoted by A B, is the set of all ordered pairs (a, b), where a A and b B. Hence,× 2 2 A B = (a, b) a A b B . × f j 2 ^ 2 g (Named after Rene Decartes) Question. Given two sets A = 1, 2, 3 and B = C, D , what is their Cartesian product A B? f g f g × A B = (1, C), (2, C), (3, C), × f (1, D), (2, D), (3, D) g p.