Discrete Mathematics

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1 Discrete Mathematics Lecture 2: Basic Structures: Set Theory MING GAO ECNU (for course related communications) Sep. 18, 2017

2 Outline 1 Set Concepts 2 Set Operations 3 Application 4 Take-aways MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

3 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 Sep. 18, / 15

4 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 Sep. 18, / 15

5 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 Sep. 18, / 15

6 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 Sep. 18, / 15

7 Set Concepts Representation of sets Tabular form A = {1, 2, 3, 4, 5}; B = { 2, 0, 2}; MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

8 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 Sep. 18, / 15

9 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 Sep. 18, / 15

10 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 Sep. 18, / 15

11 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 Sep. 18, / 15

12 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);

13 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 Sep. 18, / 15

14 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 Sep. 18, / 15

15 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};

16 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}; A B = A B; A B = A + B A B. MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

17 Set identities Set Operations Table of logical equivalence equivalence name A U = A Identity laws A = A A A = A Idempotent laws A A = A (A B) C = A (B C) Associative laws (A B) C = A (B C) A (A B) = A Absorption laws A (A B) = A A A = Complement laws A A = U MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

18 Set Operations Logical equivalence Cont d Table of logical equivalence equivalence name A U = U Domination laws A = A B = B A Commutative laws A B = B A (A B) C = (A C) (B C) Distributive laws (A B) C = (A C) (B C) (A B) = A B De Morgan s laws (A B) = A B MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

19 Proof of De Morgan law Set Operations Proof Use set builder notation and logical equivalences to establish the first De Morgan law Steps Reasons A B primise

20 Proof of De Morgan law Set Operations Proof Use set builder notation and logical equivalences to establish the first De Morgan law Steps Reasons A B primise = {x : x A B} definition of complement

21 Proof of De Morgan law Set Operations Proof Use set builder notation and logical equivalences to establish the first De Morgan law Steps Reasons A B primise = {x : x A B} definition of complement = {x : (x A B)} definition of does not belong symbol

22 Proof of De Morgan law Set Operations Proof Use set builder notation and logical equivalences to establish the first De Morgan law Steps Reasons A B primise = {x : x A B} definition of complement = {x : (x A B)} definition of does not belong symbol = {x : (x A x B)} definition of intersection

23 Proof of De Morgan law Set Operations Proof Use set builder notation and logical equivalences to establish the first De Morgan law Steps Reasons A B primise = {x : x A B} definition of complement = {x : (x A B)} definition of does not belong symbol = {x : (x A x B)} definition of intersection = {x : (x A) (x B)} De Morgan law for logic

24 Proof of De Morgan law Set Operations Proof Use set builder notation and logical equivalences to establish the first De Morgan law Steps Reasons A B primise = {x : x A B} definition of complement = {x : (x A B)} definition of does not belong symbol = {x : (x A x B)} definition of intersection = {x : (x A) (x B)} De Morgan law for logic = {x : (x A) (x B)} definition of does not belong symbol

25 Proof of De Morgan law Set Operations Proof Use set builder notation and logical equivalences to establish the first De Morgan law Steps Reasons A B primise = {x : x A B} definition of complement = {x : (x A B)} definition of does not belong symbol = {x : (x A x B)} definition of intersection = {x : (x A) (x B)} De Morgan law for logic = {x : (x A) (x B)} definition of does not belong symbol = {x : (x A) (x B)} definition of complement

26 Proof of De Morgan law Set Operations Proof Use set builder notation and logical equivalences to establish the first De Morgan law Steps Reasons A B primise = {x : x A B} definition of complement = {x : (x A B)} definition of does not belong symbol = {x : (x A x B)} definition of intersection = {x : (x A) (x B)} De Morgan law for logic = {x : (x A) (x B)} definition of does not belong symbol = {x : (x A) (x B)} definition of complement = {x : x (A B)} definition of union

27 Proof of De Morgan law Set Operations Proof Use set builder notation and logical equivalences to establish the first De Morgan law Steps Reasons A B primise = {x : x A B} definition of complement = {x : (x A B)} definition of does not belong symbol = {x : (x A x B)} definition of intersection = {x : (x A) (x B)} De Morgan law for logic = {x : (x A) (x B)} definition of does not belong symbol = {x : (x A) (x B)} definition of complement = {x : x (A B)} definition of union = A B meaning of set builder notation MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

28 Set Operations Generalized unions and intersections Union A 1 A 2 A n = n i=1 A i; A 1 A 2 A n = i=1 A i; MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

29 Set Operations Generalized unions and intersections Union A 1 A 2 A n = n i=1 A i; A 1 A 2 A n = i=1 A i; Intersection A 1 A 2 A n = n i=1 A i; A 1 A 2 A n = i=1 A i; MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

30 Set covering problem Application Input Universal set U = {u 1, u 2,, u n } Subsets S 1, S 2,, S m U Cost c 1, c 2,, c m MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

31 Set covering problem Application Input Universal set U = {u 1, u 2,, u n } Subsets S 1, S 2,, S m U Cost c 1, c 2,, c m Goal Find a set I {1, 2,, m} that minimizes i I c i, such that i I S i = U MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

32 Application Set covering problem Input Universal set U = {u 1, u 2,, u n } Subsets S 1, S 2,, S m U Cost c 1, c 2,, c m Goal Find a set I {1, 2,, m} that minimizes i I c i, such that i I S i = U Applications Document summarization Natural language generation Information cascade MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

33 Take-aways Take-aways Course homepage Advices to learning DM Not a reading course. More than a mathematics course, though it is workload-heavy MING GAO (DaSE@ecnu) Discrete Mathematics Sep. 18, / 15

CSC Discrete Math I, Spring Sets

CSC Discrete Math I, Spring Sets CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its