Sets Huynh Tuong Nguyen, Tran Huong Lan Chapter Sets Discrete Structures for Computing on 21 March 2011 Huynh Tuong Nguyen, Tran Huong Lan Faculty of Computer Science and Engineering University of Technology - VNUHCM 3.1 Contents Sets Huynh Tuong Nguyen, Tran Huong Lan 3.2 Set Definition Sets Huynh Tuong Nguyen, Tran Huong Lan • Set is a fundamental discrete structure on which all discrete structures are built • Sets are used to group objects, which often have the same properties Example • Set of all the students who are currently taking Discrete Mathematics course • Set of all the subjects that K2011 students have to take in the first semester • Set of natural numbers N Definition A set is an unordered collection of objects The objects in a set are called the elements (phần tử ) of the set A set is said to contain (chứa) its elements 3.3 Notations Sets Huynh Tuong Nguyen, Tran Huong Lan Definition • a ∈ A: a is an element of the set A • a∈ / A: a is not an element of the set A Definition (Set Description) • The set V of all vowels in English alphabet, V = {a, e, i, o, u} • Set of all real numbers greater than 1??? {x | x ∈ R, x > 1} {x | x > 1} {x : x > 1} 3.4 Equal Sets Sets Huynh Tuong Nguyen, Tran Huong Lan Definition Two sets are equal iff they have the same elements • (A = B) ↔ ∀x(x ∈ A ↔ x ∈ B) Example • {1, 3, 5} = {3, 5, 1} • {1, 3, 5} = {1, 3, 3, 3, 5, 5, 5, 5} 3.5 Venn Diagram Sets Huynh Tuong Nguyen, Tran Huong Lan • John Venn in 1881 • Universal set (tập vũ trụ) is represented by a rectangle • Circles and other geometrical figures are used to represent sets • Points are used to represent particular elements in set 3.6 Special Sets Sets Huynh Tuong Nguyen, Tran Huong Lan • Empty set (tập rỗng ) has no elements, denoted by ∅, or {} • A set with one element is called a singleton set • What is {∅}? • Answer: singleton 3.7 Subset Sets Huynh Tuong Nguyen, Tran Huong Lan Definition The set A is called a subset (tập con) of B iff every element of A is also an element of B, denoted by A ⊆ B If A = B, we write A ⊂ B and say A is a proper subset (tập thực sự) of B • ∀x(x ∈ A → x ∈ B) • For every set S, (i) ∅ ⊆ S, (ii) S ⊆ S 3.8 Cardinality Sets Huynh Tuong Nguyen, Tran Huong Lan Definition If S has exactly n distinct elements where n is non-negative integers, S is finite set (tập hữu hạn), and n is cardinality (bản số ) of S, denoted by |S| Example • A is the set of odd positive integers less than 10 |A| = • S is the letters in Vietnamese alphabet, |S| = 29 • Null set |∅| = Definition A set that is infinite if it is not finite Example • Set of positive integers is infinite 3.9 Power Set Sets Huynh Tuong Nguyen, Tran Huong Lan Definition Given a set S, the power set (tập lũy thừa) of S is the set of all subsets of the set S, denoted by P (S) Example What is the power set of {0, 1, 2}? P ({0, 1, 2}) = {∅, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}} Example • What is the power set of the empty set? • What is the power set of the set {∅} 3.10 Ordered n-tuples Sets Huynh Tuong Nguyen, Tran Huong Lan Definition The ordered n-tuple (dãy thứ tự) (a1 , a2 , , an ) is the ordered collection that has a1 as its first element, a2 as its second element, , and an as its nth element Definition Two ordered n-tuples (a1 , a2 , , an ) = (b1 , b2 , , bn ) iff = bi , for i = 1, 2, , n Example 2-tuples, or ordered pairs (cặp), (a, b) and (c, d) are equal iff a = c and b = d 3.12 Cartesian Product Sets Huynh Tuong Nguyen, Tran Huong Lan • René Descartes (1596–1650) Definition Let A and B be sets The Cartesian product (tích Đề-các) of A and B, denoted by A × B, is the set of ordered pairs (a, b), where a ∈ A and b ∈ B Hence, A × B = {(a, b) | a ∈ A ∧ b ∈ B} Example Cartesian product of A = {1, 2} and B = {a, b, c} Then A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)} Show that A × B = B × A 3.13 Sets Cartesian Product Huynh Tuong Nguyen, Tran Huong Lan Definition A1 ×A2 ×· · ·×An = {(a1 , a2 , , an ) | ∈ Ai for i = 1, 2, , n} Example A = {0, 1}, B = {1, 2}, C = {0, 1, 2} What is A × B × C? A×B×C = {(0, 1, 0), (0, 1, 1), (0, 1, 2), (0, 2, 0), (0, 2, 1), (0, 2, 2), (1, 1, 0), (1, 1, 1), (1, 1, 2), (1, 2, 0), (1, 2, 1), (1, 2, 2)} 3.14 Sets Union Huynh Tuong Nguyen, Tran Huong Lan Definition The union (hợp) of A and B A ∪ B = {x | x ∈ A ∨ x ∈ B} A∪B A B • Example: • {1,2,3} ∪ {2,4} = {1,2,3,4} • {1,2,3} ∪ ∅ = {1,2,3} 3.15 Sets Intersection Huynh Tuong Nguyen, Tran Huong Lan Definition The intersection (giao) of A and B A ∩ B = {x | x ∈ A ∧ x ∈ B} A∩B A B Example: • {1,2,3} ∩ {2,4} = {2} • {1,2,3} ∩ N = {1,2,3} 3.16 Union/Intersection Sets Huynh Tuong Nguyen, Tran Huong Lan n Ai = A1 ∪ A2 ∪ ∪ An = {x | x ∈ A1 ∨ x ∈ A2 ∨ ∨ x ∈ An } i=1 n Ai = A1 ∩ A2 ∩ ∩ An = {x | x ∈ A1 ∧ x ∈ A2 ∧ ∧ x ∈ An } i=1 3.17 Sets Difference Huynh Tuong Nguyen, Tran Huong Lan Definition The difference (hiệu) of A and B A − B = {x | x ∈ A ∧ x ∈ / B} A−B A B Example: • {1,2,3} - {2,4} = {1,3} • {1,2,3} - N = ∅ 3.18 Sets Complement Huynh Tuong Nguyen, Tran Huong Lan Definition The complement (phần bù) of A A = {x | x ∈A} / Example: • A = {1,2,3} then A = ??? • Note that A - B = A ∩ B 3.19 Sets Set Identities Huynh Tuong Nguyen, Tran Huong Lan A∪∅ A∩U = = A A Identity laws Luật đồng A∪U A∩∅ = = U ∅ Domination laws Luật nuốt A∪A A∩A = = A A Idempotent laws Luật lũy đẳng ¯ (A) = A Complementation law Luật bù 3.20 Sets Set Identities Huynh Tuong Nguyen, Tran Huong Lan A∪B A∩B = = B∪A B∩A A ∪ (B ∪ C) A ∩ (B ∩ C) = = (A ∪ B) ∪ C (A ∩ B) ∩ C Associative laws Luật kết hợp A ∪ (B ∩ C) A ∩ (B ∪ C) = = (A ∪ B) ∩ (A ∪ C) (A ∩ B) ∪ (A ∩ C) Distributive laws Luật phân phối A∪B A∩B = = A∩B A∪B Commutative laws Luật giao hoán De Morgan’s laws Luật De Morgan 3.21 Method of Proofs of Set Equations Sets Huynh Tuong Nguyen, Tran Huong Lan To prove A = B, we could use • Venn diagrams • Prove that A ⊆ B and B ⊆ A • Use membership table • Use set builder notation and logical equivalences 3.22 Example (1) Sets Huynh Tuong Nguyen, Tran Huong Lan Example Verify the distributive rule P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R) 3.23 Example (2) Sets Huynh Tuong Nguyen, Tran Huong Lan Example Prove: A ∩ B = A ∪ B (1) Show that A ∩ B ⊆ A ∪ B Suppose that x ∈ A ∩ B By the definition of complement, x ∈ / A∩B So, x ∈ / A or x ∈ /B ¯ Hence, x ∈ A¯ or x ∈ B We conclude, x ∈ A ∪ B Or, A ∩ B ⊆ A ∪ B (2) Show that A ∪ B ⊆ A ∩ B 3.24 Sets Example (3) Huynh Tuong Nguyen, Tran Huong Lan Prove: A ∩ B = A ∪ B A B A∩B A∩B ¯ A¯ ∪ B 1 0 1 0 0 1 1 1 3.25 Sets Example (4) Huynh Tuong Nguyen, Tran Huong Lan Prove: A ∩ B = A ∪ B A∩B = {x|x ∈ A ∩ B} = {x|¬(x ∈ A ∩ B)} = {x|¬(x ∈ A ∧ x ∈ B)} = {x|¬(x ∈ A) ∨ ¬(x ∈ B)} = {x|x ∈ A ∨ x ∈ B} = {x|x ∈ A ∨ x ∈ B} = {x|x ∈ A ∪ B} 3.26 ... power set has 2n elements Prove using induction! 3.11 Ordered n-tuples Sets Huynh Tuong Nguyen, Tran Huong Lan Definition The ordered n-tuple (dãy thứ tự) (a1 , a2 , , an ) is the ordered collection... and an as its nth element Definition Two ordered n-tuples (a1 , a2 , , an ) = (b1 , b2 , , bn ) iff = bi , for i = 1, 2, , n Example 2-tuples, or ordered pairs (cặp), (a, b) and (c, d)... difference (hiệu) of A and B A − B = {x | x ∈ A ∧ x ∈ / B} A−B A B Example: • {1,2,3} - {2,4} = {1,3} • {1,2,3} - N = ∅ 3.18 Sets Complement Huynh Tuong Nguyen, Tran Huong Lan Definition The complement