Set Identities the Cartesian Product Partitions Sections 3.5-3.7

Total Page:16

File Type:pdf, Size:1020Kb

Set Identities the Cartesian Product Partitions Sections 3.5-3.7 Set Identities The Cartesian Product Partitions Sections 3.5-3.7 Prof. Sandy Irani Set Operations and Logic • Set operations can be defined using logic operations: Assume all sets and elements are contained in the Universal set U (x ∈ A ∩ B) ↔ (x ∈ A) ᴧ ( x ∈ B) (x ∈ A ∪ B) ↔ (x ∈ A) v (x ∈ B) (x ∈ 퐴ഥ) ↔ ¬(x ∈ A) x ∈ ↔ F x ∈ U ↔ T Can prove set identities that mirror the laws of logic: x ∈ A ∪ ↔ (x ∈ A) v (x ∈ ) Definition of set union ↔ (x ∈ A) v F Definition of ↔ (x ∈ A) Identity Law (x ∈ A ∪ ) ↔ (x ∈ A) A ∪ = A Set Identities * Which set is not equal to Ø? You can assume that A, B, and C are not empty. A) B ∩ C ∩ 퐵ഥ B) A ∩ (B ∩ 퐵ഥ ) C) A ∩ (B ∪ 퐵ഥ ) D) (B ∪ 퐵ഥ ) Pairs, Triplets and Tuples • (a, b) is an ordered pair. – Parens (as opposed to {}) indicate that order matters: • (a, b) ≠ (b, a) • {a, b} = {b, a} • (a, b, c) is an ordered triple – b is the second entry of the triple (a, b, c) • (a, b, c, d) is an ordered 4-tuple • (a1, a2 , …, an) is an ordered n-tuple. Cartesian Product • Let S and T be sets Cartesian product of S and T is S x T = { (s, t) : s ∈ S and t ∈ T } • Example: S = {a, b, c} T = {1, 2} – S x T = { (a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2) } (a, 2) ∈ S x T ? (2, c) ∈ T x S ? a ∈ S x T ? (a, b) ∈ S x S ? (2, c) ∈ S x T ? S x T ∩ T x S ? Cartesian Plane y x ℝ x ℝ = { (x, y): x ∈ ℝ and y ∈ ℝ} Cartesian Products * S = {1, 2} T = { x, y, z } What is S x T ∩ S? A) Ø B) S C) S ∩ T D) S x T Cartesian Product on More than Two Sets • Sets A, B, and C: – A x B x C = { (a, b, c): a ∈ A and b ∈ B and c ∈ C } • Sets A1, A2 , …, An A1x A2 x … x An= { (a1, a2 , …, an): a1 ∈ A1 and a2 ∈ A2 and… and an ∈ An } Example: Drink = {OJ, Coffee} Main = {Waffles, Eggs, Pancakes} Side = {Hash browns, Toast} • Breakfast Selections = Drink x Main x Side – (OJ, Eggs, Toast) ∈ Drink x Main x Side Cartesian Product • T = {1, 2} T x T = T2 = { (1, 1), (1, 2), (2, 1), (2, 2) } T ⊆ T2 ? Let S be a set: n S = S x S x … x S = { (s1, .., sn) : each si in S, for 1 ≤ i ≤ n } • Example: {0, 1}5 • Example: ℝ4 N-tuples and Strings • If is a set of single characters, elements in n can be denoted without the punctuation, in which case they are called strings. Example: = {a, b} • (a, b, a, b) ∈ 4 (denoted as an n-tuple) • abab ∈ 4 (denoted as a string) • {0, 1}3 = set of all binary strings with 3 bits: – {0, 1}3 = { 000, 001, 010, 011, 100, 101, 110, 111 } • n-tuple punctuation is important if the underlying set is not a set of single characters! Strings • Concatenation: x = abba y = bab Concatenation of x and y is xy = abbabab Concatenation of x and a is abbaa • Empty string has no characters: – x = x = x • The length of a string x (denoted by |x|) is the number of characters in the string: Example: |abba| = 4. Cartesian Products * x = abba y = bab What is |xyλ|? A) 3 B) 4 C) 7 D) 8 Infinite sets of strings • The set of all strings of any length over an alphabet : * = 0 ∪ 1 ∪ 2 ∪ ….. Example: {0, 1}* = {, 0, 1, 00, 01, 10, 11, 000,….} • The set of all strings of any length over an alphabet : + = 1 ∪ 2 ∪ 3 ∪ ….. Example: {0, 1}+ = {0, 1, 00, 01, 10, 11, 000,….} Disjoint Sets • Two sets, A and B, are disjoint if A ∩ B = • A collection of sets A1, A2,.., An are pairwise disjoint if Ai ∩ Aj = if i j. Partition • A partition of a non-empty set A is a collection of sets A1, A2,.., An such that 1. Ai ⊆ A, for each i ∈ {1, 2,…, n} 2. A A, for each i ∈ {1, 2,…, n} i A 3. A , A ,.., A are pairwise disjoint 1 2 n l b g 4. A1 ∪ A2 ∪.. ∪ An = A i j d e c h f k a Cartesian Products * A = {0, 1}4 A1 = all 4-bit strings that start with 1 A2 = all 4-bit strings that start with 0 A3 = all 4-bit strings that start with 00 A4 = all 4-bit strings that start with 000 A5 = { 0000 } Do A1, A2, A3, A4, A5 form a partition of A? A) Yes B) No Cartesian Products * A = {0, 1}4 A1 = all 4-bit strings that start with 1 A2 = all 4-bit strings that start with 01 A3 = all 4-bit strings that start with 001 A4 = all 4-bit strings that start with 0001 Do A1, A2, A3, A4, A5 form a partition of A? A) Yes B) No.
Recommended publications
  • Lecture 7 1 Overview 2 Sequences and Cartesian Products
    COMPSCI 230: Discrete Mathematics for Computer Science February 4, 2019 Lecture 7 Lecturer: Debmalya Panigrahi Scribe: Erin Taylor 1 Overview In this lecture, we introduce sequences and Cartesian products. We also define relations and study their properties. 2 Sequences and Cartesian Products Recall that two fundamental properties of sets are the following: 1. A set does not contain multiple copies of the same element. 2. The order of elements in a set does not matter. For example, if S = f1, 4, 3g then S is also equal to f4, 1, 3g and f1, 1, 3, 4, 3g. If we remove these two properties from a set, the result is known as a sequence. Definition 1. A sequence is an ordered collection of elements where an element may repeat multiple times. An example of a sequence with three elements is (1, 4, 3). This sequence, unlike the statement above for sets, is not equal to (4, 1, 3), nor is it equal to (1, 1, 3, 4, 3). Often the following shorthand notation is used to define a sequence: 1 s = 8i 2 N i 2i Here the sequence (s1, s2, ... ) is formed by plugging i = 1, 2, ... into the expression to find s1, s2, and so on. This sequence is (1/2, 1/4, 1/8, ... ). We now define a new set operator; the result of this operator is a set whose elements are two-element sequences. Definition 2. The Cartesian product of sets A and B, denoted by A × B, is a set defined as follows: A × B = f(a, b) : a 2 A, b 2 Bg.
    [Show full text]
  • Cartesian Powers
    Cartesian Powers sequences subsets functions exponential growth +, -, x, … Analogies between number and set operations Numbers Sets Addition Disjoint union Subtraction Complement Multiplication Cartesian product Exponents ? Cartesian Powers of a Set Cartesian product of a set with itself is a Cartesian power A2 = A x A Cartesian square An ≝ A x A x … x A n’th Cartesian power n X |An| = | A x A x … x A | = |A| x |A| x … x |A| = |A|n Practical and theoretical applications Applications California License Plates California Till 1904 no registration 1905-1912 various registration formats one-time $2 fee 1913 ≤6 digits 106 = 1 million If all OK Sam? 1956 263 x 103 ≈ 17.6 m 1969 263 x 104 ≈ 176 m Binary Strings {0,1}n = { length-n binary strings } n-bit strings 0 1 1 n Set Strings Size {0,1} 0 {0,1}0 Λ 1 00 01 1 {0,1}1 0, 1 2 {0,1}2 2 {0,1}2 00, 01, 10, 11 4 000, 001, 011, 010, 10 11 3 {0,1}3 8 100, 110, 101, 111 001 011 … … … … 010 {0,1}3 n {0,1}n 0…0, …, 1…1 2n 000 111 101 | {0,1}n | =|{0,1}|n = 2n 100 110 Subsets The power set of S, denoted ℙ(S), is the collection of all subsets of S ℙ( {a,b} ) = { {}, {a}, {b}, {a,b} } ℙ({a,b}) and {0,1}2 ℙ({a,b}) a b {0,1}2 Subsets Binary strings {} 00 |ℙ(S)| = ? of S of length |S| ❌ ❌ {b} ❌ ✅ 01 {a} ✅ ❌ 10 |S| 1-1 correspondence between ℙ(S) and {0,1} {a,b} ✅ ✅ 11 |ℙ(S)| = | {0,1}|S| | = 2|S| The size of the power set is the power of the set size Functions A function from A to B maps every element a ∈ A to an element f(a) ∈ B Define a function f: specify f(a) for every a ∈ A f from {1,2,3} to {p, u} specify
    [Show full text]
  • Course 221: Michaelmas Term 2006 Section 1: Sets, Functions and Countability
    Course 221: Michaelmas Term 2006 Section 1: Sets, Functions and Countability David R. Wilkins Copyright c David R. Wilkins 2006 Contents 1 Sets, Functions and Countability 2 1.1 Sets . 2 1.2 Cartesian Products of Sets . 5 1.3 Relations . 6 1.4 Equivalence Relations . 6 1.5 Functions . 8 1.6 The Graph of a Function . 9 1.7 Functions and the Empty Set . 10 1.8 Injective, Surjective and Bijective Functions . 11 1.9 Inverse Functions . 13 1.10 Preimages . 14 1.11 Finite and Infinite Sets . 16 1.12 Countability . 17 1.13 Cartesian Products and Unions of Countable Sets . 18 1.14 Uncountable Sets . 21 1.15 Power Sets . 21 1.16 The Cantor Set . 22 1 1 Sets, Functions and Countability 1.1 Sets A set is a collection of objects; these objects are known as elements of the set. If an element x belongs to a set X then we denote this fact by writing x ∈ X. Sets with small numbers of elements can be specified by listing the elements of the set enclosed within braces. For example {a, b, c, d} is the set consisting of the elements a, b, c and d. Two sets are equal if and only if they have the same elements. The empty set ∅ is the set with no elements. Standard notations N, Z, Q, R and C are adopted for the following sets: • the set N of positive integers; • the set Z of integers; • the set Q of rational numbers; • the set R of real numbers; • the set C of complex numbers.
    [Show full text]
  • Equivalents to the Axiom of Choice and Their Uses A
    EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper.
    [Show full text]
  • Worksheet: Cardinality, Countable and Uncountable Sets
    Math 347 Worksheet: Cardinality A.J. Hildebrand Worksheet: Cardinality, Countable and Uncountable Sets • Key Tool: Bijections. • Definition: Let A and B be sets. A bijection from A to B is a function f : A ! B that is both injective and surjective. • Properties of bijections: ∗ Compositions: The composition of bijections is a bijection. ∗ Inverse functions: The inverse function of a bijection is a bijection. ∗ Symmetry: The \bijection" relation is symmetric: If there is a bijection f from A to B, then there is also a bijection g from B to A, given by the inverse of f. • Key Definitions. • Cardinality: Two sets A and B are said to have the same cardinality if there exists a bijection from A to B. • Finite sets: A set is called finite if it is empty or has the same cardinality as the set f1; 2; : : : ; ng for some n 2 N; it is called infinite otherwise. • Countable sets: A set A is called countable (or countably infinite) if it has the same cardinality as N, i.e., if there exists a bijection between A and N. Equivalently, a set A is countable if it can be enumerated in a sequence, i.e., if all of its elements can be listed as an infinite sequence a1; a2;::: . NOTE: The above definition of \countable" is the one given in the text, and it is reserved for infinite sets. Thus finite sets are not countable according to this definition. • Uncountable sets: A set is called uncountable if it is infinite and not countable. • Two famous results with memorable proofs.
    [Show full text]
  • Advanced Calculus 1. Definition: the Cartesian Product a × B
    Advanced Calculus 1. Definition: The Cartesian product A × B of sets A and B is A × B = {(a, b)| a ∈ A, b ∈ B}. This generalizes in the obvious way to higher Cartesian products: A1 × · · · × An = {(a1, . , an)| a1 ∈ A1, . , an ∈ An}. 2. Example: The real line R, the plane R2 = R×R, the n-dimensional real vector space Rn, R × [0, T], Rn × [0, T] etc. 3. Definition: The dot product (or inner or scalar product) of vectors x = (x1, . , xn) n and y = (y1, . , yn) in R is n X x · y = xiyi. i=1 n 4. Definition: The magnitude (or 2-norm) of a vector x = (x1, . , xn) in R is 1 " n # 2 X 1 2 2 |x| = xi = (x · x) . i=1 5. Definition: We write f : A 7→ B if f is a function whose domain is a subset of A and whose range is a subset of B. Let x = (x1, . , xn) n n be a point in R . So, if f(x) = f(x1, . , xn) is a function that has its domain in R , we write f : Rn 7→ R. So, for example, if 2 2 f(x1, x2, x3, x4) = x2x4 sin (x1 + x3), then f : R4 7→ R. We say that f takes R4 to R. 6. This notion is easily generalized to functions from Rn to Rm. Let x ∈ Rn and n f1(x), . , fm(x) be functions taking R to R. Then we can define a function f : Rn 7→ Rm, by f(x) = (f1(x), . , fm(x)). 2 7.
    [Show full text]
  • Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
    Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite.
    [Show full text]
  • Functions, Relations, Partial Order Relations Definition: the Cartesian Product of Two Sets a and B (Also Called the Product
    Functions, Relations, Partial Order Relations Definition: The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product) is defined to be the set of all pairs (a,b) where a ∈ A and b ∈ B . It is denoted A X B. Example : A ={1, 2), B= { a, b} A X B = { (1, a), (1, b), (2, a), (2, b)} Solve the following: X = {a, c} and Y = {a, b, e, f}. Write down the elements of: (a) X × Y (b) Y × X (c) X 2 (= X × X) (d) What could you say about two sets A and B if A × B = B × A? Definition: A function is a subset of the Cartesian product.A relation is any subset of a Cartesian product. For instance, a subset of , called a "binary relation from to ," is a collection of ordered pairs with first components from and second components from , and, in particular, a subset of is called a "relation on ." For a binary relation , one often writes to mean that is in . If (a, b) ∈ R × R , we write x R y and say that x is in relation with y. Exercise 2: A chess board’s 8 rows are labeled 1 to 8, and its 8 columns a to h. Each square of the board is described by the ordered pair (column letter, row number). (a) A knight is positioned at (d, 3). Write down its possible positions after a single move of the knight. Answer: (b) If R = {1, 2, ..., 8}, C = {a, b, ..., h}, and P = {coordinates of all squares on the chess board}, use set notation to express P in terms of R and C.
    [Show full text]
  • CARTESIAN PRODUCT Tuple Is a Collective Name for Double, Triple
    For updated version of this document see http://imechanica.org/node/19709 Zhigang Suo CARTESIAN PRODUCT Tuple is a collective name for double, triple, quadruple, etc. An n-tuple is an ordered list of n items. The Cartesian product of n sets S ,...,S is the 1 n collection of all tuples, each of which draws the first item from S ,…, and draws 1 the nth item from S . Tuples and Cartesian products are everywhere in our daily n life. Tuple Tuple. A tuple is an ordered list of items. An n-tuple is an ordered list of n items. The integer n is called the length of the tuple. We write a tuple by listing the items between parentheses, ( ) . Examples. The notation 6,1,7,4,9,5,3,7,8,9 ( ) denotes a ten-tuple, an ordered list of ten numbers. The ten-tuple happens to be a telephone number in the United States. The meaning of a tuple is understood in context. For example, we often write the three-tuple 3,0,9 as 309. The tuple can mean Room 9 on the third ( ) floor. The same tuple can also mean the number 309 in the base-10 numeral system. In this case, the tuple is shorthand for an expression: 2 1 0 309 = 3×10 + 0×10 + 9 ×10 . The items listed in a tuple, of course, need not be numbers. For example, the notation Eltz, Metternich, Reichsburg ( ) denote three castles along the Mosel River. Tuple vs. set. Tuple and set are both collections of elements, but they differ in two ways.
    [Show full text]
  • Set Theory, Relations, and Functions (II)
    LING 106. Knowledge of Meaning Lecture 2-2 Yimei Xiang Feb 1, 2017 Set theory, relations, and functions (II) Review: set theory – Principle of Extensionality – Special sets: singleton set, empty set – Ways to define a set: list notation, predicate notation, recursive rules – Relations of sets: identity, subset, powerset – Operations on sets: union, intersection, difference, complement – Venn diagram – Applications in natural language semantics: (i) categories denoting sets of individuals: common nouns, predicative ADJ, VPs, IV; (ii) quantificational determiners denote relations of sets 1 Relations 1.1 Ordered pairs and Cartesian products • The elements of a set are not ordered. To describe functions and relations we will need the notion of an ordered pair, written as xa; by, where a is the first element of the pair and b is the second. Compare: If a ‰ b, then ... ta; bu “ tb; au, but xa; by ‰ xb; ay ta; au “ ta; a; au, but xa; ay ‰ xa; a; ay • The Cartesian product of two sets A and B (written as A ˆ B) is the set of ordered pairs which take an element of A as the first member and an element of B as the second member. (1) A ˆ B “ txx; yy | x P A and y P Bu E.g. Let A “ t1; 2u, B “ ta; bu, then A ˆ B “ tx1; ay; x1; by; x2; ay; x2; byu, B ˆ A “ txa; 1y; xa; 2y; xb; 1y; xb; 2yu 1.2 Relations, domain, and range •A relation is a set of ordered pairs. For example: – Relations in math: =, ¡, ‰, ... – Relations in natural languages: the instructor of, the capital city of, ..
    [Show full text]
  • CSE541 CARDINALITIES of SETS BASIC DEFINITIONS and FACTS Cardinality Definition Sets a and B Have the Same Cardinality Iff ∃F
    CSE541 CARDINALITIES OF SETS BASIC DEFINITIONS AND FACTS 1¡1;onto Cardinality de¯nition Sets A and B have the same cardinality i® 9f( f : A ¡! B): Cardinality notations If sets A and B have the same cardinality we denote it as: jAj = jBj or cardA = cardB, or A » B. We aslo say that A and B are equipotent. Cardinality We put the above notations together in one de¯nition: 1¡1;onto jAj = jBj or cardA = cardB, or A » B i® 9f( f : A ¡! B): 1¡1;onto Finite A set A is ¯nite i® 9n 2 N 9f (f : f0; 1; 2; :::; n ¡ 1g ¡! A), i.e. we say: a set A is ¯nite i® 9n 2 N(jAj = n). In¯nite A set A is in¯nite i® A is NOT ¯nite. Cardinality Aleph zero @0 (Aleph zero) is a cardinality of N (Natural numbers). A set A has a cardinality @0 (jAj = @0) i® A » N, (or jAj = jNj, or cardA = cardN). Countable A set A is countable i® A is ¯nite or jAj = @0. In¯nitely countable A set A is in¯nitely countable i® jAj = @0. Uncountable A set A is uncountable i® A is NOT countable, Fact A set A is uncountable i® A is in¯nite and jAj 6= @0. Cardinality Continuum C (Continuum ) is a cardinality of Real Numbers, i.e. C = jRj. Sets of cardinality continuum. A set A has a cardinality C, jAj = C i® jAj = jRj, or cardA = cardR). Cardinality A · Cardinality B jAj · jBj i® A » C and C ⊆ B.
    [Show full text]
  • The Axiom of Determinacy
    Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2010 The Axiom of Determinacy Samantha Stanton Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Physical Sciences and Mathematics Commons © The Author Downloaded from https://scholarscompass.vcu.edu/etd/2189 This Thesis is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. College of Humanities and Sciences Virginia Commonwealth University This is to certify that the thesis prepared by Samantha Stanton titled “The Axiom of Determinacy” has been approved by his or her committee as satisfactory completion of the thesis requirement for the degree of Master of Science. Dr. Andrew Lewis, College of Humanities and Sciences Dr. Lon Mitchell, College of Humanities and Sciences Dr. Robert Gowdy, College of Humanities and Sciences Dr. John Berglund, Graduate Chair, Mathematics and Applied Mathematics Dr. Robert Holsworth, Dean, College of Humanities and Sciences Dr. F. Douglas Boudinot, Graduate Dean Date © Samantha Stanton 2010 All Rights Reserved The Axiom of Determinacy A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. by Samantha Stanton Master of Science Director: Dr. Andrew Lewis, Associate Professor, Department Chair Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, Virginia May 2010 ii Acknowledgment I am most appreciative of Dr. Andrew Lewis. I would like to thank him for his support, patience, and understanding through this entire process.
    [Show full text]