Speaking Mathematically Chapter 2 PDF

\ \ 2.1 THE LANGUAGE OF SETS Use of the word set as a formal mathematical term was introduced in 1879 by George Cantor (1845- 1918)  The Language of Sets Notation: If S is a set, the notation 𝑥 ∈ 𝑆, means than x is an element of S. The notation 𝑥 ∉ 𝑆 me...

\ \ 2.1 THE LANGUAGE OF SETS Use of the word set as a formal mathematical term was introduced in 1879 by George Cantor (1845- 1918)  The Language of Sets Notation: If S is a set, the notation 𝑥 ∈ 𝑆, means than x is an element of S. The notation 𝑥 ∉ 𝑆 means x is not an element of S. A set may be specified using the set-roster notation by writing all of its elements between braces. For example, {1,2,3} denote as the set whose elements are 1,2 and 3. A variation of the notation is sometimes used to describe a very large set as when we write,{1,2,3,…,100}, refers to a set of integers from 1 to 100. A similar notation can also describe an infinite set as when we write, {1,2,3,…}, refers to the set of all positive integers.  EXAMPLES Using the Set-Roster Notation 1. Let 𝐴 = 1,2,3 , 𝐵 = 2,3,1 , 1. A, B and C have exactly the same three elements: 1, 𝐶 = 1 , 1, 2, 3, 3, 3,. What are the elements of 2, and 3. Therefore A,B and C are simply different ways to represent the same set. A, B and C? How are A, B, and C related? 2. They are not equal. Because {0} is a set with one 2. Is {0} = 0? element, namely 0, whereas 0 is just zero. 3. How many elements are there in set {1, {1} }? 3. It has 2 elements: 1 and the set whose only element is 1. 4. For each nonnegative integer n, 4. 𝑈1 = 1, −1 , 𝑈2 = 2. −2 , 𝑙𝑒𝑡 𝑈𝑛 = 𝑛, −𝑛. Find 𝑈1 , 𝑈2 , 𝑎𝑛𝑑 𝑈0. 𝑈0 = 0, 0 = {0}  CHECK YOUR PROGRESS Using the Set-Roster Notation 1. Let 𝑋 = 𝑎, 𝑏, 𝑐 , 𝑌 = 𝑐, 𝑎, 𝑏 , 𝑍 = 𝑎, 𝑎, 𝑏, 𝑏, 𝑐, 𝑐, 𝑐,. What are the elements of X, Y and Z? How are X, Y, and Z related? 2. How many elements are there in set {a, {a,b}, {a} }? 3. For each positive integer x, 𝑙𝑒𝑡 𝐴𝑥 = 𝑥, 𝑥 2. Find 𝐴1 , 𝐴2 , 𝑎𝑛𝑑 𝐴3. CERTAIN SET OF NUMBERS THAT ARE FREQUENTLY USED Symbol Set ℝ Set of all real numbers ℤ Set of all integers ℚ Set of all rational numbers, or quotients of integers  SET- BUILDER NOTATION Let S denote a set and let 𝑃 𝑥 be a property that elements of S may or may not satisfy. We may define a new set to be 𝒕𝒉𝒆 𝒔𝒆𝒕 𝒐𝒇 𝒂𝒍𝒍 𝒆𝒍𝒎𝒆𝒏𝒕𝒔 𝒙 𝒊𝒏 𝑺 𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕 𝑷 𝒙 𝒊𝒔 𝒕𝒓𝒖𝒆. We denote this set as follows: {𝑥 ∈ 𝑆Τ𝑃 𝑥 }  EXAMPLES Describe the following sets Answers a. An open interval of real numbers (strictly) a. {𝑥 ∈ 𝑅| − 2 < 𝑥 < 5} between -2 and 5.It is pictured using the number line. b. {𝑥 ∈ 𝑍| − 2 < 𝑥 < 5} b. {−1, 0, 1, 2, 3, 4} c. {𝑥 ∈ 𝑍 + | − 2 < 𝑥 < 5} c. {1, 2, 3, 4}  CHECK YOUR PROGRESS Describe the following sets a. {𝑥 ∈ 𝑅| − 5 < 𝑥 < 1} b. {𝑥 ∈ 𝑍| − 1 ≤ 𝑥 < 6} c. {𝑥 ∈ 𝑍 − | − 4 ≤ 𝑥 ≤ 0}  SUBSETS If A and B are sets, then A is called a subset of B, written 𝐴 ⊆ 𝐵, if and only if, every element of A is also an element of B. 𝐴 ⊆ 𝐵 means that For all elements x, if 𝑥 ∈ 𝐴, 𝑡ℎ𝑒𝑛 𝑥 ∈ 𝐵 𝐴 ⊆ 𝐵 means that For all elements x, if 𝑥 ∈ 𝐴, 𝑡ℎ𝑒𝑛 𝑥 ∉ 𝐵 𝐴 𝑖𝑠 𝑎 𝒏𝒐𝒕 𝒂 𝒔𝒖𝒃𝒔𝒆𝒕 𝑜𝑓 𝐵 means that there is at least one element of A that is not an element of B. 𝐴 𝑖𝑠 𝒑𝒓𝒐𝒑𝒆𝒓 𝒔𝒖𝒃𝒔𝒆𝒕 𝑜𝑓 𝐵 if, and only if, every element of B is in B but there is at least one element of B that is not in A.  EXAMPLES Let 𝑨 = 𝒁+ , 𝑩 = 𝒏 ∈ 𝒁 0 ≤ 𝑛 ≤ 100 , 𝑎𝑛𝑑 Answers C = 100,200,300,400,500. a. False. Zero is not a positive integer. Thus 0 is in B, but it is not in A. 𝐵 ⊆ 𝐴 Evaluate the truth and falsity of the following statements: b. True. Each element in C is a positive integer and, hence, is in A, but there are a. 𝐵 ⊆ 𝐴 elements in A that are not in C. b. C is a proper subset of A c. C and B have at least one element in c. True. For example, 100 is in both C common and B d. 𝐶 ⊆ 𝐵 d. False. For example, 200 is in C but not e. 𝐶 ⊆ 𝐶 in B e. True. Every element in C is in C.  CHECK YOUR PROGRESS Let 𝑨 = {𝟐, 𝟐 , 𝟐)𝟐 , 𝑩 = {2, 2 , 2 }, 𝑎𝑛𝑑 𝐶 = {2} Evaluate the truth and falsity of the following statements: a. A ⊆ 𝐵 b. 𝐵 ⊆ 𝐴 c. A is a proper subset of B d. 𝐶 ⊆ 𝐵 e. C is a proper subset of A EXAMPLES Distinction between ∈ , ⊆ a. 2 ∈ {1,2,3} b. 2 ∈ 1, 2,3 c. 2 ⊆ 1, 2, 3 d. 2 ⊆ 1, 2, 3 e. 2 ⊆ 1, 2 f. 2 ∈{ 1 , 2 }  CARTESIAN PRODUCT Given sets A and B, the Cartesian product of A and B, denoted AxB read “A cross B”, is the set of all ordered pairs (a, b), where a is in A and b is in B. Symbolically: AxB ={ (a,b)/ a  A and b  B }  EXAMPLES Cartesian Products Answers Let 𝐴 = 1, 2, 3 , 𝑎𝑛𝑑 𝐵 = 𝑢, 𝑣 , 𝑓𝑖𝑛𝑑 a. AxB b. BxA c. BxB d. How many elements are in AxB, BxA, and BxB?  CHECK YOUR PROGRESS Cartesian Products: Let Y = 𝑎, 𝑏, 𝑐 , 𝑎𝑛𝑑 𝑍 = 1,2 , 𝑓𝑖𝑛𝑑 a. Y x Z b. Z x Y c. Y x Y d. How many elements are in Y x Z, Z x Y, and Y x Y?