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Chapter 4 Sets, , and

Section 4.1: Sets

CS 130 – Discrete Structures

• A set is a collection of elements or objects, and an is a member of a set – Traditionally, sets are described by capital letters, and elements by lower case letters • The  means belongs to: – Used to represent the fact that an element belongs to a particular set. – aA means that element a belongs to set A – bA implies that b is not an element of A • Braces {} are used to indicate a set • Example: A = {2, 4, 6, 8, 10} – 3A and 2A

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)[(xA  xB) Λ (xB  xA)] • Finite and : described by 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 of all positive even : – List (or partially list) its elements: • S = {2, 4, 6, 8, …} – Use 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 } • Use predicates: S = {x  P(x)} means (x)[(xS  P(x)) Λ (P(x) xS)] 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

• 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 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 – R = set of all real numbers

• A set that has no members is called a null or 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 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:

• Say S is the set of all people, M is the set of all male humans, and is the set of all computer 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 of C, however, since there are elements of M which are not in C (specifically, all males who are not studying ). – 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 of a set is simply the number of elements within the set; the cardinality of S is denoted by |S|. Then by the above 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 . • 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 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 on a set defined as follows:

–  is a 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 (, , , etc.), and – Venn diagrams.

• A rectangle marked by U is used to represent the .

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

• If A and B are subsets of S, then the 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 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- (x1, x2, …, xn) of elements of A.

• Given A = {1,2} and B = {3,4} ; find – A  B ; B  A – A2 and A3

CS 130 – Discrete Structures 22 Basic Set Identities

• Given sets A, B, and C, and a universal set U and a null/empty set , the following properties hold: – (cp) • A  B = B  A A  B = B  A – (ap) • A  (B  C) = (A  B)  C • A  (B  C) = (A  B)  C – Distributive properties (dp) • A  (B  C) = (A  B)  (A  C) • A  (B  C) = (A  B)  (A  C) – Identity properties (ip) •   A = A   = A U  A = A  U = A – Complement properties (comp) • A  A = U A  A = 

CS 130 – Discrete Structures 23 More Set Identities

• Union and Intersection with U – A  U = U A  U = A • Double Complement Law – (A) = A • Idempotent Laws – A  A = A A  A = A • Absorption properties – A  (A  B) = A A  (A  B) = A • Alternate Set Difference Representation – A - B = A  B • Inclusion in Union – A  A  B B  A  B • Inclusion in Intersection – A  B  A A  B  B • Transitive Property of Subsets – if A  B, and B  C, then A  C CS 130 – Discrete Structures 24 Exercise

• Use the set identities to prove – [A  (B  C)]  ([A  (B  C)]  (B  C)) = 

: [A  (B  C)]  ([A  (B  C)]  (B  C)) = ([A  (B  C)]  [A  (B  C)])  (B  C) using ap ([(B  C)  A]  [(B  C)  A])  (B  C) using cp twice [(B  C)  (A  A)]  (B  C) using dp [(B  C)  ]  (B  C) using comp (B  C)  (B  C) using ip  using comp

CS 130 – Discrete Structures 25