King Fahd University of Petroleum Minerals Information Computer

King Fahd University of Petroleum & Minerals Information & Computer Science Department ICS 253: Discrete Structures I Basic Structures: Sets, Functions, Sequences and Sums

ICS 253: Discrete Structures I 2 Basic Structures: Sets, Functions, Sequences and Sums Reading Assignment • K. H. Rosen, Discrete Mathematics and Its Applications, 7 th Global Ed. , Mc. Graw-Hill, 2006. • Chapter 2 (Except Section 2. 6)

ICS 253: Discrete Structures I 3 Basic Structures: Sets, Functions, Sequences and Sums Introduction • Many important discrete structures are built using sets. • For example: combinations used extensively in counting, relations, graphs and finite state machines. • Functions play important roles throughout discrete mathematics. • For example, they are used to represent the computational complexity of algorithms, to study the size of sets, to count objects, etc. • Sequences and strings are special types of functions. • We will introduce some important types of sequences, and will address the problem of identifying a pattern for the terms of a sequence from its first few terms. • Using the notion of a sequence, we will define what it means for a set to be countable. • Adding consecutive terms of a sequence, making a sum, will prove to be helpful in many discrete structures applications.

ICS 253: Discrete Structures I 4 Basic Structures: Sets, Functions, Sequences and Sums Section 2. 1: Sets • A Set is an unordered collection of “objects”. • • Defined by Cantor 1895 Objects in a set are also called elements or members of a set. Example: A set of vowels, V= {a, e, i, o, u} Bertrand Russel in 1902 showed that this definition may lead to paradoxes. • A paradox means a logical inconsistency. • Paradoxes occur if no “restriction” is made on the objects of a set • Q 29 pp. 128: Russel’s Paradox: Let S contain all sets x where x does not belong to itself, i. e. S = { x | x x}. Show that S is not well-defined by showing that both S S and S S lead to a contradiction

ICS 253: Discrete Structures I 5 Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 1(a) pp 127: List the members of these sets. a) {x | x is a real number such that x 2 = 1} • Q 2 (b, c) pp 127: Use set builder notation to give a description of each of these sets. b) {-3, -2, -1, 0, 1, 2, 3} c) {m, n, o, p}

ICS 253: Discrete Structures I 6 Basic Structures: Sets, Functions, Sequences and Sums Some Notations and Preliminaries • • • ℕ = {0, 1, 2, …} set of natural numbers ℤ = {…, -2, -1, 0, 1, 2, …} set of integers ℝ: set of real numbers ℚ: set of rational numbers Two sets are equal if and only if they have the same elements. • i. e. order and repetitions are irrelevant. • The set A is a subset of B if and only if every element of A is also an element of B, denoted by A B. • • i. e. x(x A x B) Prove that S for all sets S

ICS 253: Discrete Structures I 7 Basic Structures: Sets, Functions, Sequences and Sums Venn Diagrams • Used to graphically represent sets • Universal set is represented by a rectangle, all other subsets are represented by circles and/or other geometric shapes. • Q#11 pp. 127: Use a Venn diagram to illustrate the relationship A B and B C.

8 ICS 253: Discrete Structures I Basic Structures: Sets, Functions, Sequences and Sums More Preliminaries • Theorem 1: For every set S • S S Proof: • Proper Subset

ICS 253: Discrete Structures I 9 Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 6 pp 127: For each of the following sets, determine whether 2 is an element of that set. a) {x ℝ | x is an integer greater than 1} b) {x ℝ | x is the square of an integer} c) {2, {2}} d) {{2}, {{2}}} e) {{2}, {2}}} f) {{{2}}}

ICS 253: Discrete Structures I 10 Basic Structures: Sets, Functions, Sequences and Sums More Preliminaries • Definition: Let S be a set. If there are exactly n distinct elements in S, where n is a nonnegative integer, we say that S is a finite set of cardinality n, denoted by |S|=n. Otherwise, the set is infinite. • What is the cardinality of the set of vowels in the English language? • Given a set S, the power set of S is the set of all subsets of the set S, and is denoted by P(S). • If |S|=n, |P(S)|=2 n elements

ICS 253: Discrete Structures I 11 Basic Structures: Sets, Functions, Sequences and Sums Examples • What is the power set of • • {1, 2} { } {{1, 2}} • What is the cardinality of each of the following sets • • {a} {{a}} {a, a, a, a} {a, {a}, {a}}}

12 ICS 253: Discrete Structures I Basic Structures: Sets, Functions, Sequences and Sums Cartesian Products • • The ordered n-tuple (a 1, a 2, …, an) is the ordered collection that has a 1 as its first element, a 2 as its second element, …, an as its nth element. Two ordered tuples (a 1, a 2, …, am) and (b 1, b 2, …, bn) are said to be equal if and only if 1. m = n and 2. ai= bi for 1 i n. • • An ordered 2 -tuple is called an ordered pair. The Cartesian product of the sets A 1, A 2, …, An, denoted by A 1 A 2 … An is the set of ordered n-tuples (a 1, a 2, …, an), where ai belongs to Ai, for i=1, 2, …, n.

ICS 253: Discrete Structures I 13 Basic Structures: Sets, Functions, Sequences and Sums Examples • What is the Cartesian product of A={1, 2}, B={3, 4} and C={5}? • Q 19 pp 128: Let A={a, b, c, d} and B={y, z}. Find a) A B b) B A

ICS 253: Discrete Structures I 14 Basic Structures: Sets, Functions, Sequences and Sums Using Set Notation with Quantifiers • x S (P(x)) is shorthand for x(x S P(x)). • Similarly, x S (P(x)) is shorthand for …………… Truth Sets of Quantifiers • Note that x P(x) is true over the domain U if and only if the truth set of P is the set U. • Likewise, x P(x) is true over the domain U if and only if the truth set of P is nonempty.

ICS 253: Discrete Structures I 15 Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 26 pp 128: Translate each of these quantifications into English and determine its truth value. a) x ℝ (x 2 – 1) b) x ℤ (x 2 = 2) c) x ℤ (x 2 > 0) d) x ℝ (x 2 = x)

ICS 253: Discrete Structures I 16 Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 27 pp 128: Find the truth set of each of these predicates where the domain is the set of integers, ℤ: a) P(x): “x 2 < 3” b) Q(x): “x 2 > x” c) R(x): “ 2 x + 1 = 0”

ICS 253: Discrete Structures I 17 Basic Structures: Sets, Functions, Sequences and Sums Section 2. 2: Set Operations • Let A and B be sets. • • The union of the sets A and B, denoted by A B, is the set that contains those elements that are either in A or in B, or in both. The intersection of the sets A and B, denoted by A B, is the set containing those elements in both A and B are called disjoint sets if their intersection is the empty set. The difference of A and B, denoted by A B, is the set containing those elements that are in A but not in B. • The difference of A and B is also called the complement of B with respect to A.

ICS 253: Discrete Structures I 18 Basic Structures: Sets, Functions, Sequences and Sums Complement of a Set • Let U be the universal set. The complement of the set A, denoted by A, is the complement of A with respect to U, i. e. U A. • Question 2 page 138: Let A={a, b, c, d, e} and B={a, b, c, d, e, f, g, h}. Find: • • A B B A

ICS 253: Discrete Structures I 19 Basic Structures: Sets, Functions, Sequences and Sums Example • Q 18 pp 138: The symmetric difference of A and B, denoted by A B, is the set containing those elements in either A or B, but not in both A and B. Find the symmetric difference of { 1, 3, 5} and {1, 2, 3}.

ICS 253: Discrete Structures I 20 Basic Structures: Sets, Functions, Sequences and Sums Cardinality of Some Set Operations • Given finite sets A and B, |A B| = |A| + |B| – |A B|. • Can you come up with a law for the |A – B|?

ICS 253: Discrete Structures I 21 Basic Structures: Sets, Functions, Sequences and Sums Set Identities Identity Laws Domination Laws Idempotent Laws Complementation Law Commutative Laws

ICS 253: Discrete Structures I 22 Basic Structures: Sets, Functions, Sequences and Sums Set Identities (Cont. ) Associative Laws Distributive Laws De Morgan’s Laws Absorption Laws Complement Laws

ICS 253: Discrete Structures I 23 Basic Structures: Sets, Functions, Sequences and Sums Set Identities Verification • Prove that • Using the definitions • Using membership tables

24 ICS 253: Discrete Structures I Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 16 pp 138: Can we conclude that A = B if A, B and C are sets such that 1. A C = B C ? 2. A C = B C ? 3. A C = B C and A C = B C?

ICS 253: Discrete Structures I 25 Basic Structures: Sets, Functions, Sequences and Sums Generalized Union and Intersection • The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection. • The intersection of a collection of sets is the set that contains those elements that are members of all the sets in the collection.

ICS 253: Discrete Structures I 26 Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 29 pp 139: Let Ai be the set of all nonempty bit strings (i. e. bit strings of length at least one) of length not exceeding i. Find 1. 2.

ICS 253: Discrete Structures I 27 Basic Structures: Sets, Functions, Sequences and Sums Computer Representation of Sets • Although sets are unordered, representing the universal set in a specific order in computers has a lot of advantages • U must be finite, with number of elements not exceeding available memory • The members of U are given an arbitrary order, i. e. {a 1, a 2, …, an} • Any subset A of U is represented with a n-bit string S, where n=|U|, such that for each element e U at position j: • If e A then Sj=1 else Sj=0 • • What is the representation of U and ? What is the intersection, union, difference?

ICS 253: Discrete Structures I 28 Basic Structures: Sets, Functions, Sequences and Sums Example • Q 35 (a, c) pp 139: Show bitwise operations on bit strings can be used to find these combinations of A = {a, b, c, d, e}, B={b, c, d, g, p, t, v}, C = {c, e, i, 0, u, x , y, z} and D = {d, e, h, i, n, o, t, u, x, y}. a) A B c) (A D) (B C) *) A B

ICS 253: Discrete Structures I 29 Basic Structures: Sets, Functions, Sequences and Sums Section 2. 3: Functions • • The concept of a function is important in discrete mathematics • Sequences and strings • Algorithm efficiency in space and time • Algorithm development through recursive functions Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A • f(a) = b • • • b is the image of a and a is the pre-image of b. f : A B (f maps A to B) A: Domain of f. B: Codomain of f. Range of f: Set of all images of elements in A. Functions are sometimes called mappings or transformations

ICS 253: Discrete Structures I 30 Basic Structures: Sets, Functions, Sequences and Sums Examples

ICS 253: Discrete Structures I 31 Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 4 pp 153: Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function. a) the function that assigns to each nonnegative integer its last digit b) the function that assigns the next largest integer to a positive integer c) the function that assigns to a bit string the number of one bits in the string d) the function that assigns to a bit string the number of bits in the string

ICS 253: Discrete Structures I 32 Basic Structures: Sets, Functions, Sequences and Sums Examples • The domain and codomain of functions are often specified in programming languages. For instance, the Java statement int floor(float real) {. . . } specifies that the domain and range of the function floor are………………

ICS 253: Discrete Structures I 33 Basic Structures: Sets, Functions, Sequences and Sums Some Operations on Functions • Let f 1 and f 2 be functions from A to ℝ. Then • f 1 + f 2 is a function from A to ℝ. • f 1 f 2 is a function from A to ℝ. • Is f 1/f 2 a function? • Let f be a function from set A to set B and let S be a subset of A. The image of S is a subset of B that consists of the images of the elements of S, denoted by f(S).

ICS 253: Discrete Structures I 34 Basic Structures: Sets, Functions, Sequences and Sums Some Functional Properties • A function f is said to be one-to-one or injective if and only if f(x)=f(y) implies that x=y for all x and y in the domain of f. The function is said to be an injection. • A function f from A to B is said to be onto or surjective if and only if for every element b B there is an element a A with f(a)=b. The function is said to be a surjection. • A function f is called a one-to-one correspondence, or a bijection, if it is both one-to-one and onto.

ICS 253: Discrete Structures I 35 Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 6 and 7 pp 153: Determine whether each of these functions from {a, b, c, d} to itself is one-to-one or onto. a) f(a) = b, f(b) = a, f(c) = c, f(d) = d b) f(a) = b, f(b) = b, f(c) = d, f(d) = c c) f(a) = d, f(b) = b, f(c) = c, f(d) = d • Q 8 and 9 pp 153: Determine whether each of these functions from ℤ to ℤ is one-to-one or onto. a) f(n) = n – 1 c) f(n) = n 3 b) f(n) = n 2 + 1 d) f(n) = n/2

ICS 253: Discrete Structures I 36 Basic Structures: Sets, Functions, Sequences and Sums More Properties • A function f whose domain and co-domain are subsets of ℝ is called strictly increasing if f(x) < f(y) whenever x < y and x and y are in the domain of f. • A function f whose domain and co-domain are subsets of ℝ is called strictly decreasing if f(x) > f(y) whenever x < y and x and y are in the domain of f.

37 ICS 253: Discrete Structures I Basic Structures: Sets, Functions, Sequences and Sums Inverse Functions • Let f be a 1: 1 correspondence from the set A onto the set B. The inverse function of f, denoted by f -1, is the function that assigns to an element b B the unique element a A such that f(a)=b. • f – 1(b)=a when f(a) = b • Find the inverse function for each 1: 1 correspondence in the previous slide.

ICS 253: Discrete Structures I 38 Basic Structures: Sets, Functions, Sequences and Sums Composition of Functions • Let g be a function from the set A to the set B and let f be a function from the set B to the set C. The composition of the functions f and g, denoted by f g is defined by (f g)(a) = f (g(a))

ICS 253: Discrete Structures I 39 Basic Structures: Sets, Functions, Sequences and Sums Example • Q 22 pp 154: Find f g and g f, where f(x)=x 2 + 1 and g(x)=x + 2, are functions from ℝ to ℝ.

ICS 253: Discrete Structures I 40 Basic Structures: Sets, Functions, Sequences and Sums Graphs of Functions • Let f be a function from the set A to the set B. The graph of the function f is the set of ordered pairs {(a, b) | a A and f(a) = b}

ICS 253: Discrete Structures I 41 Basic Structures: Sets, Functions, Sequences and Sums Graph of f(n)=1 – n 2 from ℤ to ℤ

ICS 253: Discrete Structures I 42 Basic Structures: Sets, Functions, Sequences and Sums Some Important Functions • The floor function assigns to the real number x the largest integer that is less than or equal to x, denoted by x. • The ceiling function assigns to the real number x the smallest integer that is greater than or equal to x, denoted by x. • 1/2 = 1/2 = • -1/2 = -1/2 = • The factorial function f: ℕ ℤ+, denoted by f(n) = n!, is the product of the first n positive integers, so f(n) = n (n – 1) … (2)(1) and f(0)=1.

ICS 253: Discrete Structures I 43 Basic Structures: Sets, Functions, Sequences and Sums Graph of f(x)= x for x in ℝ

ICS 253: Discrete Structures I 44 Basic Structures: Sets, Functions, Sequences and Sums Graph of f(x)= x for x in ℝ

ICS 253: Discrete Structures I 45 Basic Structures: Sets, Functions, Sequences and Sums Graph of f(x)= x/2 for x in ℝ

ICS 253: Discrete Structures I 46 Basic Structures: Sets, Functions, Sequences and Sums Useful Properties of the Floor and Ceiling Functions

ICS 253: Discrete Structures I 47 Basic Structures: Sets, Functions, Sequences and Sums Example • Prove that x + n = x + n, where x ℝ and n ℤ

ICS 253: Discrete Structures I 48 Basic Structures: Sets, Functions, Sequences and Sums Partial Functions •

ICS 253: Discrete Structures I 49 Basic Structures: Sets, Functions, Sequences and Sums Section 2. 4: Sequences and Summations • A sequence is a function from a subset of the set of integers (usually either the set {0, 1, 2, . . . } or the set {1, 2, 3, . . . }) to a set S. • We use the notation an to denote the image of the integer n. • We call an a term of the sequence. • The notation {an} is used to describe the sequence

ICS 253: Discrete Structures I 50 Basic Structures: Sets, Functions, Sequences and Sums Notation • A geometric progression is a sequence of the form a, ar 2 , . . . , ar n , . . . where the initial term a and the common ratio r are real numbers. • A geometric progression is a discrete analogue of the exponential function f (x) = ar x.

ICS 253: Discrete Structures I 51 Basic Structures: Sets, Functions, Sequences and Sums Notation • An arithmetic progression is a sequence of the form a, a + d, a + 2 d, . . . , a + n d, . . . where the initial term a and the common difference d are real numbers. • An arithmetic progression is a discrete analogue of the linear function ……

ICS 253: Discrete Structures I 52 Basic Structures: Sets, Functions, Sequences and Sums Examples • What is the term a 8 of the sequence {an} if an equals a) 2 n – l? b) 7? c) 1 + (– 1)n ? d) –(– 2)n?

ICS 253: Discrete Structures I 53 Basic Structures: Sets, Functions, Sequences and Sums Recurrence Relations • A recurrence relation for the sequence {an} is an equation that expresses an in terms of one or more of the previous terms of the sequence, namely, a 0, a 1, . . . , an− 1, for all integers n with n ≥ n 0, where n 0 is a nonnegative integer. • A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation.

ICS 253: Discrete Structures I 54 Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 6 pp 167: Find the first five terms of the sequence defined by each of these recurrence relations and initial conditions. a) an = 6 an – 1, a 0 = 2 c) an = an – 1 + 3 an – 2, a 0 = 1, al = 2 e) an = an – 1 + an – 3, a 0 = 1, al = 2, a 2 = 0

ICS 253: Discrete Structures I 55 Basic Structures: Sets, Functions, Sequences and Sums Examples • Q 8 pp 167: Show that the sequence {an} is a solution of the recurrence relation an = – 3 an – 1 + 4 an – 2 if a) an = 0. b) an = 1. c) an = (– 4)n. d) an = 2(– 4)n + 3.

ICS 253: Discrete Structures I 56 Basic Structures: Sets, Functions, Sequences and Sums Examples • Suppose that a person deposits $10, 000 in a savings account at a bank yielding 11% per year with interest compounded annually. How much will be in the account after 30 years?

ICS 253: Discrete Structures I 57 Basic Structures: Sets, Functions, Sequences and Sums Examples • Find a recurrence relation and give initial conditions for the number of bit strings of length n that do not have two consecutive 0 s. How many such bit strings are there of length five?

ICS 253: Discrete Structures I 58 Basic Structures: Sets, Functions, Sequences and Sums Sequence Generalization • The problem is how to generalize a sequence from its first few terms. • Examples • • • 1, 1/2, 1/4, 1/8, 1/16, … 1, 3, 5, 7, 9, … 1, – 1, 1, – 1 , 1, … 1, 2, 2, 3, 3, 3, 4, 4, … 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, …

ICS 253: Discrete Structures I 59 Basic Structures: Sets, Functions, Sequences and Sums A Table to Memorize! nth term First 10 terms n 2 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, . . . n 3 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, . . . n 4 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, . . . 2 n 2, 4, 8, 16, 32, 64, 128, 256 , 512, 1024, . . . 3 n 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, . . . n! 1, 2, 6, 24, 120, 720, 5040, 40320, 3628800, . . .

ICS 253: Discrete Structures I 60 Basic Structures: Sets, Functions, Sequences and Sums More Examples • For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. a) 1, 7, 25, 79, 241, 727, 2185, 6559, 19681, 59047, …

ICS 253: Discrete Structures I 61 Basic Structures: Sets, Functions, Sequences and Sums b) 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, . . . c) 1, 10, 11, 100, 101, 110, 111, 1000, 1001 , 1010, 1011, . . .

ICS 253: Discrete Structures I 62 Basic Structures: Sets, Functions, Sequences and Sums Summations j : index of summation, can be replaced by any arbitrary variable m: lower limit n: upper limit Important rule:

ICS 253: Discrete Structures I 63 Basic Structures: Sets, Functions, Sequences and Sums Examples • Express the sum of the first 100 terms of the sequence {an}, where an = 1/n for n = 1, 2, 3, … • •

ICS 253: Discrete Structures I 64 Basic Structures: Sets, Functions, Sequences and Sums Index Changes in the Summation • Consider the summation and assume that we want the index to start from 0 to n – 1 rather than 1 to n. How do we change the index?

ICS 253: Discrete Structures I 65 Basic Structures: Sets, Functions, Sequences and Sums Theorem 1 • If a and r are real numbers and r 0, then Proof

ICS 253: Discrete Structures I 66 Basic Structures: Sets, Functions, Sequences and Sums Examples • •

ICS 253: Discrete Structures I 67 Basic Structures: Sets, Functions, Sequences and Sums Some Useful Summations

ICS 253: Discrete Structures I 68 Basic Structures: Sets, Functions, Sequences and Sums More Examples • Find • Let x be a real number with |x|<1. Find • Find

ICS 253: Discrete Structures I 69 Basic Structures: Sets, Functions, Sequences and Sums More Examples • For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, . . .

ICS 253: Discrete Structures I 70 Basic Structures: Sets, Functions, Sequences and Sums Cardinality • Definition: The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B. • Definition: A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. When an infinite set S is countable, we denote the cardinality of 0 S by (where 0 is aleph, the first letter of the Hebrew alphabet). We write |S| 0 and say that S has = � cardinality "aleph null. "

ICS 253: Discrete Structures I 71 Basic Structures: Sets, Functions, Sequences and Sums Cardinality • Question: What do we need to do to find whether a set is countable or not? • Example 1: Show that the set of odd positive integers is a countable set.

ICS 253: Discrete Structures I 72 Basic Structures: Sets, Functions, Sequences and Sums Cardinality • Example 2: Show that the set of all integers is countable.

ICS 253: Discrete Structures I 73 Basic Structures: Sets, Functions, Sequences and Sums Cardinality • Example 3: Show that the set of positive rational numbers is countable.

ICS 253: Discrete Structures I 74 Basic Structures: Sets, Functions, Sequences and Sums Cardinality • Example 4: Show that the set of real numbers is an uncountable set.

ICS 253: Discrete Structures I 75 Basic Structures: Sets, Functions, Sequences and Sums Reading • Hilbert’s Grand Hotel