Chapter 4 Sets, Combinatorics, and Probability Section 4.1: Sets CS 130 – Discrete Structures Set Theory • A set is a collection of elements or objects, and an element is a member of a set – Traditionally, sets are described by capital letters, and elements by lower case letters • The symbol means belongs to: – Used to represent the fact that an element belongs to a particular set. – aA means that element a belongs to set A – bA implies that b is not an element of A • Braces {} are used to indicate a set • Example: A = {2, 4, 6, 8, 10} – 3A and 2A CS 130 – Discrete Structures 2 Continue on Sets • Ordering is not imposed on the set elements and listing elements twice or more is redundant • Two sets are equal if and only if they contain the same elements – A = B means (x)[(xA xB) Λ (xB xA)] • Finite and infinite set: described by number of elements – Members of infinite sets cannot be listed but a pattern for listing elements could be indicated CS 130 – Discrete Structures 3 Various Ways To Describe A Set • The set S of all positive even integers: – List (or partially list) its elements: • S = {2, 4, 6, 8, …} – Use recursion to describe how to generate the set elements: • 2 S, if n S, then (n+2) S – Describe a property P that characterizes the set elements • in words: S = {x | x is a positive even integer} • Use predicates: S = {x P(x)} means (x)[(xS P(x)) Λ (P(x) xS)] where P is the unary predicate. • Hence, every element of S has the property P and everything that has a property P is an element of S. CS 130 – Discrete Structures 4 Examples • Describe each of the following sets by listing the elements: – {x | x is a month with exactly thirty days} – {x | x is an integer and 4<x<9} • What is the predicate/property for each of the following sets: – {1, 4, 9, 16, …..} – {2, 3, 5, 7, 11, 13, 17, ….} CS 130 – Discrete Structures 5 Standard Sets • Some notations used for convenience of defining sets – N = set of all nonnegative integers • remember that 0 N – Z = set of all integers – Q = set of all rational numbers – R = set of all real numbers • A set that has no members is called a null or empty set and is represented by – S = {x | x N and x < 0} – Note that is not the same as {} CS 130 – Discrete Structures 6 Examples • What are the elements in the following sets? – A = {x| (y)(y {0,1,2} and x = y2} – B = {x| x N and (y) (y N and x <= y)} – C = {x| x N and (y) (y N x <= y)} – D = {x| (y)(z)(y{1,2} and z{2,3} and x=z–y)} CS 130 – Discrete Structures 7 Open and Closed Interval • {x R | -2 < x < 3} – denotes the set containing all real numbers between -2 and 3. This is an open interval, meaning that the endpoints -2 and 3 are not included. By all real numbers, we mean everything; 1.05, -3/4, and every other real number within that interval. • {x R | -2 x 3} – is a closed interval -- it includes all the numbers in the open interval described above, plus the endpoints. CS 130 – Discrete Structures 8 Relationships Between Sets: Subsets • Say S is the set of all people, M is the set of all male humans, and C is the set of all computer science students: – M and C are both subsets of S, because all elements of M and C are also elements of S. – M is not a subset of C, however, since there are elements of M which are not in C (specifically, all males who are not studying computer science). – For sets S and M, M is a subset of S if and only if, every element in M is also an element of S. – Symbolically: M S (x) (x M x S). – If M S and M S, then there is at least one element of S that is not an element of M, then M is a proper subset of S denoted by M S CS 130 – Discrete Structures 9 More Relationships • A superset is the opposite of subset. If M is a subset of S, then S is a superset of M, denoted S M. Likewise, if M is a proper subset of S, then S is a proper superset of M, denoted S M. • The cardinality of a set is simply the number of elements within the set; the cardinality of S is denoted by |S|. Then by the above definition of subset, it is clear that set M must have fewer members than S, which yields the following symbolic representation: – M S |M| < |S| – M S |M| <= |S| CS 130 – Discrete Structures 10 Exercises A = {x| x N and x 5} B = {10, 12, 16, 20} C = {x | (y)(y N and x = 2y)} What statements are true? • B C B A • A C 26 C • {11, 12, 13} A {11, 12, 13} C • {12} B {12} B • {x | x N and x < 20} B 5 A • {} B A For the following sets, prove A B • A = { x | x R such that x2 – 4x + 3 = 0} • B = { x | x N and 1 x 4} CS 130 – Discrete Structures 11 Set of Sets • From every set, many subsets can be generated. A set whose elements are all such subsets is called the power set. • For a set S, (S) is termed as the power set. • For a set S = {1, 2, 3} ; – (S) = { ,{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} • For a set with n elements, the power set has 2n elements. CS 130 – Discrete Structures 12 Set Operations • Binary and unary operators – Binary acts on two elements – Unary acts on a single element • An ordered pair of elements is written as (x,y) and is different from (y,x) • Two ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d • If S = {2,3}, the ordered pairs of this set are (2,2), (2,3), (3,2), (3,3) CS 130 – Discrete Structures 13 Binary Operations • Binary operation on a set defined as follows: – is a binary operation if for every ordered pair (x,y) of elements of S, x y exists, and is unique, and is a member of S. – The fact that x y exists and is unique is the same as saying that the binary operation is well-defined. – The fact that x y always belongs to S means that S is closed under the operation • The operator is just a placeholder for the real operator like +, -, *, etc. CS 130 – Discrete Structures 14 Examples • Example: +, - and * are all binary operators on Z • How about division(/)? Is it a binary operation on Z and why? • Is subtraction a binary operation on N? • Let x^ = -x, so that x^ is the negative of x. – Is ^ a unary operation on Z? – Is ^ a unary operation on N? CS 130 – Discrete Structures 15 More Examples • Which of the following are neither binary nor unary operations on the given sets? • Why? x y = x/y; S = set of all positive integers x y = x/y; S = set of all positive rational numbers y x y = x ; S = R x y = maximum of x and y; S = N x y = x + y; S = set of Fibonacci numbers CS 130 – Discrete Structures 16 Operations On Sets • New sets can be formed in a variety of ways, and can be described using: – Set builder notation (union, intersection, complement, etc.), and – Venn diagrams. • A rectangle marked by U is used to represent the universal set. CS 130 – Discrete Structures 17 Union and Intersection • The union of set A and B is the set which contains all elements in either set A or set B. – A B = {x | x A or x B } • The intersection of set A and B is the set which contains all elements both in set A and in set B. – A B = {x | x A and x B } • Example – A = {1, 3, 5, 7, 9} – B = {1, 2, 3, 4, 5} CS 130 – Discrete Structures 18 Disjoint, Universal and Difference Sets • Given set A and set B, if A B = , then A and B are disjoint sets. In other words, there are no elements in A which are also in B. • The complement of set A, denoted as ~A or A, is the set of all elements that are not in A. – A = {x | x A } • The difference of A-B, is the set of elements in A which are not in B. This is also known as the complement of B relative to A. – A - B = {x | x A and x B } – Note that A – B = A B B – A CS 130 – Discrete Structures 19 Exercise Let A = {1, 2, 3, 5, 10} B = {2, 4, 7, 8, 9} C = {5, 8, 10} be subsets of S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find – A B = {1, 2, 3, 4, 5, 7, 8, 9, 10} – A – C = {1, 2, 3} – B (A C) = {1, 3, 5, 6, 10} {1, 2, 3, 5, 8, 10} = {1, 3, 5, 10} – A B C = {} – (A B) C = {1, 2, 3, 4, 7, 9} CS 130 – Discrete Structures 20 Cartesian Product • If A and B are subsets of S, then the Cartesian product or cross product of A and B denoted symbolically by A B is defined by: – A B = {(x,y) | x A and y B } • The Cartesian product of 2 sets is the set of all combinations of ordered pairs • For example, given 2 sets A and B, where A = {a, b, c} and B = {1, 2, 3}, the Cartesian product of A and B can be represented as: – A B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)} – Is A B = B A ? CS 130 – Discrete Structures 21 Continue On Cartesian Product • Cross-product of a set with itself is represented as A A or A2 n • A represents the set of all n-tuples (x1, x2, …, xn) of elements of A.