Basic Structures: Sets, Functions, Sequences, Sums, & Matrices Chapter 2 PDF

Basic Structures: Sets, Functions, Sequences, Sums, & Matrices Chapter 2 With Question/Answer Animations  Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Funct...

Basic Structures: Sets, Functions, Sequences, Sums, & Matrices Chapter 2 With Question/Answer Animations  Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability Sequences & Summations Types of Sequences Summation Formulae Set Cardinality Countable Sets Sets Section 2.1  Section Summary Definition of sets Describing Sets Roster Method Set-Builder Notation Some Important Sets in Math Empty Set & Universal Set Subsets & Set Equality Cardinality of Sets Tuples Cartesian Product  Introduction Sets are one of the basic building blocks for the types of objects considered in discrete math. Important for counting. Programming languages have set operations. Set theory is an important branch of math. A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to The notation a ∈ A denotes that a is an element of the set A. contain its elements. If a is not a member of A, write a ∉ A  Describing a Set: Roster Method S = {a,b,c,d} Order not important S = {a,b,c,d} = {b,c,a,d} Each distinct object is either a member or not; listing more than once does not change the set. S = {a,b,c,d} = {a,b,c,b,c,d} Ellipses (…) may be used to describe a set without listing all of the members when the pattern is clear. S = {a,b,c,d, ……,z } e.g., Set of all vowels in the English alphabet: V = {a,e,i,o,u} Set of all odd positive integers less than 10: O = {1,3,5,7,9} Set of all positive integers less than 100: S = {1,2,3,……..,99} Set of all integers less than 0: S = {…., -3,-2,-1}  Some Important Sets N = natural numbers = {0,1,2,3….} Z = integers = {…,-3,-2,-1,0,1,2,3,…} Z⁺ = positive integers = {1,2,3,…..} R = set of real numbers R+ = set of positive real numbers C = set of complex numbers. Q = set of rational numbers  Set-Builder Notation Specify the property or properties that all members must satisfy: S = {x | x is a positive integer less than 100} O = {x | x is an odd positive integer less than 10} O = {x ∈ Z⁺ | x is odd & x < 10} A predicate may be used: S = {x | P(x)} e.g., S = {x | Prime(x)} Positive rational numbers: Q+ = {x ∈ R | x = p/q, for some positive integers p,q}  Interval Notation [a,b] = {x | a ≤ x ≤ b} [a,b) = {x | a ≤ x < b} (a,b] = {x | a < x ≤ b} (a,b) = {x | a < x < b} closed interval [a,b] open interval (a,b)  Universal Set & Empty Set The universal set U is the set containing everything currently under consideration. Contents depend on the context. Venn Diagram The empty set is the set with no U elements. Symbolized ∅, but {} also used. V aei ou John Venn (1834-1923) Cambridge, UK Russell’s Paradox Let S be the set of all sets which are not members of themselves. A paradox results from trying to answer the question “Is S a member of itself?” Related Paradox: Henry is a barber who shaves all people who do not shave themselves. A paradox results from trying to answer the question “Does Henry shave himself?” Bertrand Russell (1872- 1970) Cambridge, UK Nobel Prize Winner  Some things to remember Sets can be elements of sets. {{1,2,3},a, {b,c}} {N,Z,Q,R} The empty set is different from a set containing the empty set. {}=∅ ≠{∅}={{ }}  Set Equality Definition: 2 sets are equal if & only if they have the same elements. Therefore if A & B are sets, then A & B are equal if & only if We write A = B if A & B are equal sets. {1,3,5} = {3, 5, 1} {1,5,5,5,3,3,1} = {1,3,5}  Subsets Definition: The set A is a subset of B, if & only if every element The notation A ⊆ B is used to indicate that A is a subset of the set B. of A is also an element of B. A ⊆ B holds if & only if 1. Because a ∈ ∅ is always false, ∅ ⊆ S ,for every set S. is true. 2. Because a ∈ S → a ∈ S, S ⊆ S, for every set S.  Showing a Set is a Subset of Another Set Showing that A is a Subset of B: To show that A ⊆ B, show that if x belongs to A, then x also belongs to B. subset of B, A ⊈ B, find an element x ∈ A with x ∉ B. (Such an x is a Showing that A is not a Subset of B: To show that A is not a counterexample to the claim that x ∈ A implies x ∈ B.) e.g., 1. The set of all computer science majors at your school is a subset of all students at your school. 2. The set of integers with squares less than 100 is not a subset of the set of nonnegative integers. Another look at Equality of Sets Recall that 2 sets A & B are equal, denoted by A= B, iff Using logical equivalences we have that A = B iff This is equivalent to A⊆B & B⊆A  Proper Subsets Definition: If A ⊆ B, but A ≠B, then we say A is a proper subset of B, denoted by A ⊂ B. If A ⊂ B, then is true. Venn Diagram U B A  Set Cardinality Definition: If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is finite. Otherwise it is infinite. Definition: The cardinality of a finite set A, denoted by |A|, is the number of (distinct) elements of A. 1. |ø| = 0 e.g., 2. Let S be the letters of the English alphabet. Then |S| = 26 3. |{1,2,3}| = 3 4. |{ø}| = 1 5. The set of integers is infinite.  Power Sets Definition: The set of all subsets of a set A, denoted P(A), is called the power set of A. e.g., If A = {a,b} then P(A) = {ø, {a},{b},{a,b}} 2ⁿ. (In Chapters 5 & 6, we will discuss different ways to show If a set has n elements, then the cardinality of the power set is this.)  Tuples The ordered n-tuple (a1,a2,…..,an) is the ordered collection that has a1 as its first element & a2 as its second element & so on until an as its last element. Two n-tuples are equal if & only if their corresponding elements are equal. 2-tuples are called ordered pairs. The ordered pairs (a,b) & (c,d) are equal if & only if a = c & b = d.  Cartesian Product Definition: The Cartesian Product of 2 sets A & B, denoted by A × B is the set of ordered pairs (a,b) where a ∈ A & b∈B. e.g., A = {a,b} B = {1,2,3} A × B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)} Definition: A subset R of the Cartesian product A × B is called a relation from the set A to the set B. Cartesian Product Definition: The cartesian products of the sets A1,A2,……,An, denoted by A1 × A2 × …… × An , is the set of ordered tuples (a1,a2,……,an) where ai belongs to Ai n- 1, … n. for i = Example: What is A × B × C where A = {0,1}, B = {1,2} and C Solution: A × B × C = {(0,1,0), (0,1,1), (0,1,2),(0,2,0), (0,2,1), = {0,1,2} (0,2,2),(1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)} Truth Sets of Quantifiers Given a predicate P and a domain D, we define the truth set of P to be the set of elements in D for which P(x) is true. The truth set of P(x) is denoted by integers and P(x) is “|x| = 1” is the set {-1,1} Example: The truth set of P(x) where the domain is the Set Operations Section 2.2  Section Summary Set Operations Union Intersection Complementation Difference More on Set Cardinality Set Identities Proving Identities Membership Tables  Boolean Algebra system called a Boolean Algebra. This is discussed in Chapter 12. Propositional calculus & set theory are both instances of an algebraic The operators in set theory are analogous to the corresponding operator in propositional calculus. As always there must be a universal set U. All sets are assumed to be subsets of U.  Union denoted by A ∪ B, is the set: Definition: Let A & B be sets. The union of the sets A & B, e.g., What is {1,2,3} ∪ {3, 4, 5}? Venn Diagram for A ∪ Solution: {1,2,3,4,5} B U A B  Intersection Definition: The intersection of sets A & B, denoted by A ∩ B, is Note if the intersection is empty, then A & B are said to be e.g., What is? {1,2,3} ∩ {3,4,5} ? disjoint. Venn Diagram for A ∩B Solution: {3} e.g.,What is? U {1,2,3} ∩ {4,5,6} ? A Solution: ∅ B  Complement respect to U), denoted by Ā is the set U - A Definition: If A is a set, then the complement of the A (with Ā = {x ∈ U | x ∉ A} (The complement of A is sometimes denoted by Ac.) complement of {x | x > 70} e.g., If U is the positive integers less than 100, what is the Solution: {x | x ≤ 70} Venn Diagram for Complement U Ā A  Difference Definition: Let A & B be sets. The difference of A & B, denoted by A – B, is the set containing the elements of A that are not in B. The difference of A & B is also called the complement of B with respect A – B = {x | x ∈ A  x ∉ B} = A ∩B to A. U Venn Diagram for A − B A B  The Cardinality of the Union of 2 Sets Inclusion-Exclusion |A ∪ B| = |A| + | B| − |A ∩ B| U A B Venn Diagram for A, B, A ∩ B, A ∪B e.g., Let A be the math majors in your class & B be the CS majors. To count the number of students who are either math majors or CS majors, add the number of math majors & the number of CS majors, & subtract the number of joint CS/math majors. We will return to this principle in Chapter 6 & Chapter 8 where we will derive a formula for the cardinality of the union of n sets, where n is a positive integer.  Review Questions e.g., U = {0,1,2,3,4,5,6,7,8,9,10} A = {1,2,3,4,5}, B ={4,5,6,7,8} 1. A ∪ B 2. A ∩ B Solution: {1,2,3,4,5,6,7,8} Solution: {4,5} 3. Ā 4. Solution: {0,6,7,8,9,10} 5. A – B Solution: {0,1,2,3,9,10} 6. B – A Solution: {1,2,3} Solution: {6,7,8}  Set Identities Identity laws Domination laws Idempotent laws Complementation law Commutative laws Associative laws Distributive laws De Morgan’s laws Absorption laws Continued on next slide  Complement laws Membership Table Exampl Construct a membership table to show that the e: distributive law holds. Solutio n: A B C 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 Functions Section 2.3  Section Summary Definition of a Function. Domain, Codomain Image, Preimage Injection, Surjection, Bijection Inverse Function Function Composition Graphing Functions Floor, Ceiling, Factorial Partial Functions (optional)  Functions Definition: Let A & B be nonempty sets. A function f from A to B, denoted f: A → B is an assignment of each element of A to exactly one element of B. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A. Functions are sometimes called mappings or Students Grade s A transformations. Carlota Rodriguez B Sandeep Patel C Jalen Williams D F Kathy Scott  Functions Given a function f: A → B: We say f maps A to B or f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, a is called the preimage of b. then b is called the image of a under f. The range of f is the set of all images of points in A under f. We denote it by f(A). Two functions are equal when they have the same domain, the same codomain & map each element of the domain to the same element of the codomain.  Representing Functions Functions may be specified in different ways: An explicit statement of the assignment. Students & grades example. A formula. f(x) = x + 1 A computer program. Fibonacci Number (covered in the next section & also inChapter 5). A Java program that when given an integer n, produces the nth  Questions f(a) = ? z A B The image of d is ? z a x The domain of f is ? A b y The codomain of f is ? B c The preimage of y is ? b d z f(A) = ? {y,z} The preimage(s) of z is (are) ? {a,c,d}  Question on Functions & Sets If & S is a subset of A, then A B f {a,b,c,} {y,z} a x is ? b f {c,d} is ? {z y } c d z  Injections Definition: A function f is said to be 1-to-1, or injective, if & only if f(a) = f(b) implies that a = b for all a & b in the domain of f. A function is said to be an injection if it is 1-to-1. A B a x v b y c z d w  Surjections Definition: A function f from A to B is called onto or surjective, if & only if for every element there is an element with. A function f is called a surjection if it is onto. A B a x b y c z d  Bijections Definition: A function f is a 1-to-1 correspondence, or a bijection, if it is both 1-to-1 & onto (surjective & injective). A a B x b y c d z w  Showing that f is 1-to-1 or onto Example 1: Let f be the function from {a,b,c,d} to {1,2,3} defined by f(a) = 3, f(b) = 2, f(c) = 1, & f(d) = 3. Is f an onto function? Solution: Yes, f is onto since all 3 elements of the codomain are images of elements in the domain. If the codomain were changed to {1,2,3,4}, f would not be onto. Example 2: Is the function f(x) = x2 from the set of integers to the set of integers onto? Solution: No, f is not onto because there is no integer x with x2 = −1, for example.  Inverse Functions Definition: Let f be a bijection from A to B. Then the inverse of f, denoted , is the function from B to A defined as No inverse exists unless f is a bijection. Why?  Inverse Functions A f B A B a V V a b b W W c c d X X d Y Y  Questions  Example 1: Let f be the function from {a,b,c} to {1,2,3} such that f(a) = 2, f(b) = 3, & f(c) = 1. Is f invertible & if so what is its inverse? Solution: The function f is invertible because it is a 1-to-1 correspondence. The inverse function f-1 reverses the correspondence given by f, so f-1 (1) = c, f-1 (2) = a, & f-1 (3) =b  Example 2: Let f: Z  Z be such that f(x) = x + 1. Is f invertible, & if so, what is its inverse? Solution: The function f is invertible because it is a 1-to-1 correspondence. The inverse function f-1 reverses the correspondence so f-1 (y) = y – 1.  Example 3: Let f: R → R be such that. Is f invertible, & if so, what is its inverse? Solution: The function f is not invertible because it is not 1-to-1. Composition Definition: Let f: B → C, g: A → B. The composition of f with g, denoted is the function from A to C defined by Composition g f A B C A C V a a h h b i b W i c c X j d d j Y Composition Example 1: If and then and Composition Questions Example 2: Let g be the function from the set {a,b,c} to itself f(a) = 3, f(b) such that g(a) = b, g(b) = c, and g(c) = a. Let f be the function = 2, and f(c) = 1. from the set {a,b,c} to the set {1,2,3} such that What is the composition of f and g, and what is the composition Solution: The composition f∘g is defined by of g and f. f∘g (a)= f(g(a)) = f(b) = 2. f∘g (b)= f(g(b)) = f(c) = 1. f∘g (c)= f(g(c)) = f(a) = 3. Note that g∘f is not defined, because the range of f is not a subset of the domain of g. Composition Questions Example 2: Let f and g be functions from the set of integers to the set of integers defined by f(x) = 2x + 3 and g(x) = 3x + 2. What is the composition of f and g, and also the composition of g and f ? f∘g (x)= f(g(x)) = f(3x + 2) = 2(3x + 2) + 3 = 6x + 7 Solution: g∘f (x)= g(f(x)) = g(2x + 3) = 3(2x + 3) + 2 = 6x + 11  Graphs of Functions function f is the set of ordered pairs {(a,b) | a ∈A & f(a) = b}. Let f be a function from the set A to the set B. The graph of the Graph of f(n) = 2n + 1 Graph of f(x) = x2 from Z to Z from Z to Z  Some Important Functions The floor function, denoted is the largest integer less than or equal to x. The ceiling function, denoted is the smallest integer greater than or equal to x e.g., Floor & Ceiling Functions Floor & Ceiling Functions  Factorial Function Definition: f: N → Z+ , denoted by f(n) = n! is the product of the first n positive integers when n is a nonnegative integer. f(n) = 1 ∙ 2 ∙∙∙ (n – 1) ∙ n, f(0) = 0! = 1 e.g., f(1) = 1! = 1 f(2) = 2! = 1 ∙ 2 = 2 Stirling’s Formula: f(6) = 6! = 1 ∙ 2 ∙ 3∙ 4∙ 5 ∙ 6 = 720 f(20) = 2,432,902,008,176,640,000. Sequences & Summations Section 2.4  Section Summary Sequences. e.g., Geometric Progression, Arithmetic Progression Recurrence Relations e.g., Fibonacci Sequence Summations  Introduction Sequences are ordered lists of elements. 1, 2, 3, 5, 8 1, 3, 9, 27, 81, ……. Sequences are important in: math, computer science. We will introduce the terminology to represent sequences & sums of the terms in the sequences.  Sequences either the set {0, 1, 2, 3, 4, …..} or {1, 2, 3, 4, ….} ) to a set S. Definition: A sequence is a function from a subset of the integers (usually The notation an is used to denote the image of the integer n. We can think of an as the equivalent of f(n) where f is a function from {0,1,2, …..} to S. We call an a term of the sequence. e.g., Consider the sequence where  Geometric Progression Definition: A geometric progression is a sequence of the form: where the initial term a & the common ratio r are real numbers. Let a = 1 & r = −1. Then: e.g., 1. 2. Let a = 2 & r = 5. Then: 3. Let a = 6 & r = 1/3. Then:  Arithmetic Progression Definition: A arithmetic progression is a sequence of the form: where the initial term a & the common difference d are real numbers. 1. Let a = −1 & d = 4: e.g., 2. Let a = 7 & d = −3: 3. Let a = 1 & d = 2:  Strings Definition: A string is a finite sequence of characters from a finite set (an alphabet). Sequences of characters or bits are important in computer science. The empty string is represented by λ. The string abcde has length 5.  Recurrence Relations Definition: A recurrence relation for the sequence {an} is an equation that expresses an in terms of 1 or more of the previous terms of the sequence, namely, a0, a1, …, an-1, for all integers n with n ≥ n0, where n0 is a nonnegative integer. A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation. The initial conditions for a sequence specify the terms that precede the first term where the recurrence relation takes effect.  Questions about Recurrence Relations Example 1: Let {an} be a sequence that satisfies the recurrence relation: an = an-1 + 3 for n = 1,2,3,4,…. & suppose that a0 = 2. What are a1 , a2 & a3? [Here a0 = 2 is the initial condition.] a1 = a 0 + 3 = 2 + 3 = 5 Solution: We see from the recurrence relation that a2 = 5 + 3 = 8 a3 = 8 + 3 = 11  Questions about Recurrence Relations Example 2: Let {an} be a sequence that satisfies the recurrence relation: an = an-1 – an-2 for n = 2,3,4,…. & suppose that a0 = 3 & a1 = 5. What are a2 & a3? [Here the initial conditions are a0 = 3 & a1 = 5. ] Solution: We see from the recurrence relation that a2 = a1 - a0 = 5 – 3 = 2 a3 = a2 – a1 = 2 – 5 = –3  Fibonacci Sequence Definition: Define the Fibonacci sequence, f0 ,f1 ,f2,…, by: Initial Conditions: f0 = 0, f1 = 1 Recurrence Relation: fn = fn-1 + fn-2 e.g., Find f2 ,f3 ,f4 , f5 & f6. Answer: f2 = f1 + f0 = 1 + 0 = 1 , f3 = f2 + f1 = 1 + 1 = 2 , f4 = f3 + f2 = 2 + 1 = 3 , f5 = f4 + f3 = 3 + 2 = 5 , f6 = f5 + f4 = 5 + 3 = 8.  Solving Recurrence Relations Finding a formula for the nth term of the sequence generated by a recurrence relation is called solving the recurrence relation. Such a formula is called a closed formula. Various methods for solving recurrence relations will be covered in Chapter 8 where recurrence relations will be studied in greater depth. Here we illustrate by example the method of iteration in which we need to guess the formula. The guess can be proved correct by the method of induction (Chapter 5).  Iterative Solution Example Method 1: Working upward, forward substitution Let {an} be a sequence that satisfies the recurrence relation an = an-1 + 3 for n = 2,3,4,…. & suppose that a1 = 2. a2 = 2 + 3 a3 = (2 + 3) + 3 = 2 + 3 ∙ 2 a4 = (2 + 2 ∙ 3) + 3 = 2 + 3 ∙ 3... an = an-1 + 3 = (2 + 3 ∙ (n – 2)) + 3 = 2 + 3(n – 1) Useful Sequences  Summations Sum of the terms from the sequence The notation: represents The variable j is called the index of summation. It runs through all the integers starting with its lower limit m & ending with its upper limit n. More generally for a set S: e.g., Geometric Series Sums of terms of geometric progressions Cardinality of Sets Section 2.5  Section Summary Cardinality Countable Sets Computability  Cardinality Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted |A| = |B|, |A| ≤ |B|. If there is a 1-to-1 function (i.e., an injection) from A to B, the cardinality of A is less than or the same as the cardinality of B & we write Definition: A set countable is either: finite or has the same cardinality as the set of positive integers (Z+) A set that is not countable is uncountable. The set of real numbers R is an uncountable set. When an infinite set is countable (countably infinite) its cardinality is ℵ0 (ℵ reads as “aleph” or “‫)”أ‬. We write |S| = ℵ0 & say that S has cardinality “aleph null.”  Showing that a Set is Countable An infinite set is countable if & only if it is possible to list the elements of the set in a sequence (indexed by the positive integers). The reason for this is that a 1-to-1 correspondence f from the set of positive integers to a set S can be expressed in terms of a sequence a1,a2,…, an ,… where a1 = f(1), a2 = f(2),…, an = f(n),… Example 1: Show that the set of positive even integers E is countable set. Solution: Let f(x) = 2x. 1 2 3 4 5 6 ….. 2 4 6 8 10 12 …… f(n) = f(m). Then 2n = 2m, & so n = m. Then f is a bijection from Z+ to E since f is both 1-to-1 & onto. To see that it is onto, suppose that t is an even positive integer. Then t = 2k for some positive To show that it is 1-to-1, suppose that integer k & f(k) = t. Showing that a Set is Countable Example 2: Show that the set of integers Z is countable. 0, 1, − 1, 2, − 2, 3, − 3 ,……….. Solution: Can list in a sequence: Or can define a bijection from Z+ to Z: When n is even: f(n) = −(n−1)/2 f(n) = n/2 When n is odd:  Class activity Q1: Let {an} be a sequence that satisfies the recurrence relation: an = an-1 +2 an-2 for n = 2,3,4,…. & suppose that a0 = 5 & a1 = 6. What are a2, a3, & a4? A B Q2: is this relation: a x a) 1-to-1 b b) Onto y c c) Bijections z d) All d w Matrices Section 2.6 Section Summary Definition of a Matrix Matrix Arithmetic Transposes and Powers of Arithmetic Zero-One matrices Matrices Matrices are useful discrete structures that can be used in many ways. For example, they are used to: describe certain types of functions known as linear transformations. Express which vertices of a graph are connected by edges (see Chapter 10). In later chapters, we will see matrices used to build models of: Transportation systems. Communication networks. Algorithms based on matrix models will be presented in later chapters. Here we cover the aspect of matrix arithmetic that will be needed later. Matrix A matrix with m rows and n columns is called an m n Definition: A matrix is a rectangular array of numbers. matrix. The plural of matrix is matrices. A matrix with the same number of rows as columns is called square. Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal. 3 2 matrix Notation Let m and n be positive integers and let The ith row of A is the 1 n matrix [ai1, ai2,…,ain]. The jth column of A is the m 1 matrix: The (i,j)th element or entry of A is the element aij. We can use A = [aij ] to denote the matrix with its (i,j)th element equal to aij. Matrix Arithmetic: Addition Defintion: Let A = [aij] and B = [bij] be m n matrices. The sum of A and B, denoted by A + B, is the m n matrix that has aij + bij as its (i,j)th element. In other words, A + B = [aij + bij]. Example: Note that matrices of different sizes can not be added. Matrix Multiplication Definition: Let A be an m k matrix and B be a k n matrix. m n matrix that has its (i,j)th element equal to the sum of the products of The product of A and B, denoted by AB, is the the corresponding elements from the ith row of A and the jth column of B. In other words, if AB = [cij] then cij = ai1b1j + ai2b2j + … + aikbkj. Example: The product of two matrices is undefined when the number of columns in the first matrix is not the same as the number of rows in the second. Illustration of Matrix Multiplication The Product of A = [aij] and B = [bij] Transpose matrix