Discrete Mathematics and Its Applications Lecture 2: Basic Structures: Set Theory MING GAO DaSE@ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Mar. 27, 2020
Outline Set Concepts 1 Set Operations 2 Application 3 Take-aways 4 MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 2 / 16
Set Concepts Set Definition Set A is a collection of objects (or elements). a ∈ A : “ a is an element of A ” or “ a is a member of A ”; a �∈ A : “ a is not an element of A ”; A = { a 1 , a 2 , · · · , a n } : A contains a 1 , a 2 , · · · , a n ; Order of elements is meaningless; It does not matter how often the same element is listed. MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 3 / 16
Set Concepts Set Definition Set A is a collection of objects (or elements). a ∈ A : “ a is an element of A ” or “ a is a member of A ”; a �∈ A : “ a is not an element of A ”; A = { a 1 , a 2 , · · · , a n } : A contains a 1 , a 2 , · · · , a n ; Order of elements is meaningless; It does not matter how often the same element is listed. Set equality Sets A and B are equal if and only if they contain exactly the same elements. If A = { 9 , 2 , 7 , − 3 } and B = { 7 , 9 , 2 , − 3 } , then A = B . If A = { 9 , 2 , 7 } and B = { 7 , 9 , 2 , − 3 } , then A � = B .; If A = { 9 , 2 , − 3 , 9 , 7 , − 3 } and B = { 7 , 9 , 2 , − 3 } , then A = B . MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 3 / 16
Set Concepts Applications Examples Bag of words model: documents, reviews, tweets, news, etc; Transactions: shopping list, app downloading, book reading, video watching, music listening, etc; Records in a DB, data item in a data streaming, etc; Neighbors of a vertex in a graph; MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 4 / 16
Set Concepts Applications Examples Bag of words model: documents, reviews, tweets, news, etc; Transactions: shopping list, app downloading, book reading, video watching, music listening, etc; Records in a DB, data item in a data streaming, etc; Neighbors of a vertex in a graph; “Standard” sets Natural numbers: N = { 0 , 1 , 2 , 3 , · · · } Integers: Z = {· · · , − 2 , − 1 , 0 , 1 , 2 , · · · } Positive integers: Z + = { 1 , 2 , 3 , 4 , · · · } Real Numbers: R = { 47 . 3 , − 12 , − 0 . 3 , · · · } Rational Numbers: Q = { 1 . 5 , 2 . 6 , − 3 . 8 , 15 , · · · } MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 4 / 16
Set Concepts Representation of sets Tabular form A = { 1 , 2 , 3 , 4 , 5 } ; B = {− 2 , 0 , 2 } ; MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 5 / 16
Set Concepts Representation of sets Tabular form A = { 1 , 2 , 3 , 4 , 5 } ; B = {− 2 , 0 , 2 } ; Descriptive form A = set of first five natural numbers; B = set of positive odd integers; MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 5 / 16
Set Concepts Representation of sets Tabular form A = { 1 , 2 , 3 , 4 , 5 } ; B = {− 2 , 0 , 2 } ; Descriptive form A = set of first five natural numbers; B = set of positive odd integers; Set builder form Q = { a / b : a ∈ Z ∧ b ∈ Z ∧ b � = 0 } ; B = { y : P ( y ) } , where P ( Y ) : y ∈ E ∧ 0 < y ≤ 50; MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 5 / 16
Set Concepts Representation of sets Remarks A = ∅ : empty set, or null set; Universal set U : contains all the objects under consideration. A = {{ a , b } , { b , c , d }} ; MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 6 / 16
Set Concepts Representation of sets Remarks A = ∅ : empty set, or null set; Universal set U : contains all the objects under consideration. A = {{ a , b } , { b , c , d }} ; Venn diagrams In general, a universal set is represented by a rectangle. MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 6 / 16
Set Concepts Subsets Definition Set A is a subset of B iff every element of A is also an element of B , denoted as A ⊆ B .
Set Concepts Subsets Definition Set A is a subset of B iff every element of A is also an element of B , denoted as A ⊆ B . A ⊆ B = ∀ x ( x ∈ A → x ∈ B );
Set Concepts Subsets Definition Set A is a subset of B iff every element of A is also an element of B , denoted as A ⊆ B . A ⊆ B = ∀ x ( x ∈ A → x ∈ B ); For every set S , we have: (1) ∅ ⊆ S ; and (2) S ⊆ S ;
Set Concepts Subsets Definition Set A is a subset of B iff every element of A is also an element of B , denoted as A ⊆ B . A ⊆ B = ∀ x ( x ∈ A → x ∈ B ); For every set S , we have: (1) ∅ ⊆ S ; and (2) S ⊆ S ; When we wish to emphasize that set A is a subset of set B but that A � = B , we write A ⊂ B and say that A is a proper subset of B , i.e., ∀ x ( x ∈ A → x ∈ B ) ∧ ∃ ( x ∈ B ∧ x �∈ A );
Set Concepts Subsets Definition Set A is a subset of B iff every element of A is also an element of B , denoted as A ⊆ B . A ⊆ B = ∀ x ( x ∈ A → x ∈ B ); For every set S , we have: (1) ∅ ⊆ S ; and (2) S ⊆ S ; When we wish to emphasize that set A is a subset of set B but that A � = B , we write A ⊂ B and say that A is a proper subset of B , i.e., ∀ x ( x ∈ A → x ∈ B ) ∧ ∃ ( x ∈ B ∧ x �∈ A ); Two useful rules: (1) A = B ⇔ ( A ⊆ B ) ∧ ( B ⊆ A ); (2) ( A ⊆ B ) ∧ ( B ⊆ C ) ⇒ ( A ⊆ C );
Set Concepts Subsets Definition Set A is a subset of B iff every element of A is also an element of B , denoted as A ⊆ B . A ⊆ B = ∀ x ( x ∈ A → x ∈ B ); For every set S , we have: (1) ∅ ⊆ S ; and (2) S ⊆ S ; When we wish to emphasize that set A is a subset of set B but that A � = B , we write A ⊂ B and say that A is a proper subset of B , i.e., ∀ x ( x ∈ A → x ∈ B ) ∧ ∃ ( x ∈ B ∧ x �∈ A ); Two useful rules: (1) A = B ⇔ ( A ⊆ B ) ∧ ( B ⊆ A ); (2) ( A ⊆ B ) ∧ ( B ⊆ C ) ⇒ ( A ⊆ C );
Set Concepts Subsets Definition Set A is a subset of B iff every element of A is also an element of B , denoted as A ⊆ B . A ⊆ B = ∀ x ( x ∈ A → x ∈ B ); For every set S , we have: (1) ∅ ⊆ S ; and (2) S ⊆ S ; When we wish to emphasize that set A is a subset of set B but that A � = B , we write A ⊂ B and say that A is a proper subset of B , i.e., ∀ x ( x ∈ A → x ∈ B ) ∧ ∃ ( x ∈ B ∧ x �∈ A ); Two useful rules: (1) A = B ⇔ ( A ⊆ B ) ∧ ( B ⊆ A ); (2) ( A ⊆ B ) ∧ ( B ⊆ C ) ⇒ ( A ⊆ C ); Given a set S , the power set of S is the set of all subsets of S , denoted as P ( S ). The size of 2 | S | , where | S | is the size of S . MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 7 / 16
Set Concepts Cartesian product Definition Let A and B be sets. The Cartesian product of A and B , denoted by A × B , is set A × B = { ( a , b ) : a ∈ A ∧ b ∈ B } , where ( a , b ) is a ordered 2-tuples, called ordered pairs.
Set Concepts Cartesian product Definition Let A and B be sets. The Cartesian product of A and B , denoted by A × B , is set A × B = { ( a , b ) : a ∈ A ∧ b ∈ B } , where ( a , b ) is a ordered 2-tuples, called ordered pairs. Let A = { 1 , 2 } and B = { a , b , c } , the Cartesian product A × B = { (1 , a ) , (1 , b ) , (1 , c ) , (2 , a ) , (2 , b ) , (2 , c ) } ;
Set Concepts Cartesian product Definition Let A and B be sets. The Cartesian product of A and B , denoted by A × B , is set A × B = { ( a , b ) : a ∈ A ∧ b ∈ B } , where ( a , b ) is a ordered 2-tuples, called ordered pairs. Let A = { 1 , 2 } and B = { a , b , c } , the Cartesian product A × B = { (1 , a ) , (1 , b ) , (1 , c ) , (2 , a ) , (2 , b ) , (2 , c ) } ; The Cartesian product of A 1 , A 2 , · · · , A n is denoted as A 1 × A 2 × · · · × A n A 1 × A 2 × · · · × A n = { ( a 1 , a 2 , · · · , a n ) : ∀ i a i ∈ A i } ;
Set Concepts Cartesian product Definition Let A and B be sets. The Cartesian product of A and B , denoted by A × B , is set A × B = { ( a , b ) : a ∈ A ∧ b ∈ B } , where ( a , b ) is a ordered 2-tuples, called ordered pairs. Let A = { 1 , 2 } and B = { a , b , c } , the Cartesian product A × B = { (1 , a ) , (1 , b ) , (1 , c ) , (2 , a ) , (2 , b ) , (2 , c ) } ; The Cartesian product of A 1 , A 2 , · · · , A n is denoted as A 1 × A 2 × · · · × A n A 1 × A 2 × · · · × A n = { ( a 1 , a 2 , · · · , a n ) : ∀ i a i ∈ A i } ; A subset R of the Cartesian product A × B is called a relation from A to B . MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 8 / 16
Set Operations Set operations Operators Let A and B be two sets, and U be the universal set
Set Operations Set operations Operators Let A and B be two sets, and U be the universal set Union: A ∪ B = { x : x ∈ A ∨ x ∈ B } ; Intersection: A ∩ B = { x : x ∈ A ∧ x ∈ B } ; Difference: A − B = { x : x ∈ A ∧ x �∈ B } (sometimes denoted as A \ B ); Complement: A = U − A = { x ∈ U : x �∈ A } ;
Set Operations Set operations Operators Let A and B be two sets, and U be the universal set Union: A ∪ B = { x : x ∈ A ∨ x ∈ B } ; Intersection: A ∩ B = { x : x ∈ A ∧ x ∈ B } ; Difference: A − B = { x : x ∈ A ∧ x �∈ B } (sometimes denoted as A \ B ); Complement: A = U − A = { x ∈ U : x �∈ A } ;
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