Sets Sets Huynh Tuong Nguyen, Tran Vinh Tan Contents Sets Set Operation 3 1 Chapter 3 Sets Discrete Structures for Computing on 21 March 2011 Huynh Tuong Nguyen, Tran Vinh Tan Faculty of Computer Scie[.]
Sets Huynh Tuong Nguyen, Tran Vinh Tan Chapter Sets Discrete Structures for Computing on 21 March 2011 Contents Sets Set Operation Huynh Tuong Nguyen, Tran Vinh Tan Faculty of Computer Science and Engineering University of Technology - VNUHCM 3.1 Sets Contents Huynh Tuong Nguyen, Tran Vinh Tan Contents Sets Sets Set Operation Set Operation 3.2 Sets Set Definition Huynh Tuong Nguyen, Tran Vinh Tan • 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 Contents Example Sets Set Operation • 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 Sets Notations Huynh Tuong Nguyen, Tran Vinh Tan Definition • a ∈ A: a is an element of the set A Contents • a∈ / A: a is not an element of the set A Sets Set Operation 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 Sets Equal Sets Huynh Tuong Nguyen, Tran Vinh Tan Definition Two sets are equal iff they have the same elements Contents Sets • (A = B) ↔ ∀x(x ∈ A ↔ x ∈ B) Set Operation Example • {1, 3, 5} = {3, 5, 1} • {1, 3, 5} = {1, 3, 3, 3, 5, 5, 5, 5} 3.5 Sets Venn Diagram Huynh Tuong Nguyen, Tran Vinh Tan • John Venn in 1881 • Universal set (tập vũ trụ) is represented by a rectangle Contents Sets Set Operation • Circles and other geometrical figures are used to represent sets • Points are used to represent particular elements in set 3.6 Sets Special Sets Huynh Tuong Nguyen, Tran Vinh Tan Contents • Empty set (tập rỗng ) has no elements, denoted by ∅, or {} • A set with one element is called a singleton set Sets Set Operation • What is {∅}? • Answer: singleton 3.7 Sets Subset Huynh Tuong Nguyen, Tran Vinh Tan 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 6= B, we write A ⊂ B and say A is a proper subset (tập thực sự) of B Contents Sets Set Operation • ∀x(x ∈ A → x ∈ B) • For every set S, (i) ∅ ⊆ S, (ii) S ⊆ S 3.8 Sets Cardinality Huynh Tuong Nguyen, Tran Vinh Tan 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| Contents Example Sets Set Operation • 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 Sets Power Set Huynh Tuong Nguyen, Tran Vinh Tan 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) Contents Sets Example Set Operation 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 Sets Ordered n-tuples Huynh Tuong Nguyen, Tran Vinh Tan 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 Contents Sets Set Operation 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 Sets Cartesian Product Huynh Tuong Nguyen, Tran Vinh Tan • 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, Contents Sets Set Operation 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 6= B × A 3.13 Sets Cartesian Product Huynh Tuong Nguyen, Tran Vinh Tan Definition A1 ×A2 ×· · ·×An = {(a1 , a2 , , an ) | ∈ Ai for i = 1, 2, , n} Contents Sets Set Operation 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 Vinh Tan Definition The union (hợp) of A and B A ∪ B = {x | x ∈ A ∨ x ∈ B} Contents Sets Set Operation 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 Vinh Tan Definition The intersection (giao) of A and B A ∩ B = {x | x ∈ A ∧ x ∈ B} Contents Sets Set Operation A∩B A B Example: • {1,2,3} ∩ {2,4} = {2} • {1,2,3} ∩ N = {1,2,3} 3.16 Sets Union/Intersection Huynh Tuong Nguyen, Tran Vinh Tan Contents n [ Sets Ai = A1 ∪ A2 ∪ ∪ An = {x | x ∈ A1 ∨ x ∈ A2 ∨ ∨ x ∈ An } Set Operation i=1 n \ Ai = A1 ∩ A2 ∩ ∩ An = {x | x ∈ A1 ∧ x ∈ A2 ∧ ∧ x ∈ An } i=1 3.17 Sets Difference Huynh Tuong Nguyen, Tran Vinh Tan Definition The difference (hiệu) of A and B A − B = {x | x ∈ A ∧ x ∈ / B} Contents Sets Set Operation A−B A B Example: • {1,2,3} - {2,4} = {1,3} • {1,2,3} - N = ∅ 3.18 Sets Complement Huynh Tuong Nguyen, Tran Vinh Tan Definition The complement (phần bù) of A A = {x | x ∈A} / Contents Sets Set Operation Example: • A = {1,2,3} then A = ??? • Note that A - B = A ∩ B 3.19 Sets Set Identities Huynh Tuong Nguyen, Tran Vinh Tan 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 Contents Sets Set Operation 3.20