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Chapter 2 Basic Concepts of Set Theory 2.1 Symbols and Terminology Set: a collection of objects Elements (members): objects belonging to a set There are three ways to designate a set: 1. Word description 2. Listing method 3. Set-builder notation Ex: 1. The set of even counting numbers less than ten 2. { 2, 4, 6, 8 } 3. { x | x is an even counting number } We use CAPITAL letters to denotes names of sets usually A, B, and C. We use lower case letters for elements of sets. Ex: A = { a, b, c} Empty Set (null set): is the set containing no elements. Notation: { } or are ways of denoting the empty set { } is not the empty set {0} also is not the empty set Sets of Real Numbers Natural (counting) numbers: { 1, 2, 3, 4, 5, …} Whole numbers: { 0, 1, 2, 3, 4, 5, …} Integers: { …-3, -2, -1, 0, 1, 2, 3, …} Rational numbers: any number that can be written in the form a b if a and b are intergers and b 0. Ex: { 5, -4, ¾, .25, .3333…. } Irrational numbers: number that aren’t rational. IE: numbers that can not be expressed as fractions. Ex: { , e, 2 , 4 8 } _________________________________________________________________________________________________________________ Cardinal Number (cardinality): the number of elements in a set; repeated elements should not be counted more than once. Notation: n(A) = “n of A” = is the cardinal number of set A. Finite Set: if the cardinal number of the set can be expressed as a whole number. Ex: {1,2,3,4,a,b,c} Infinite Set: if the cardinal number of a set can not be expressed as a whole number. Ex: The set of counting numbers Symbol Notation :is and element of :is not an element of Ex: A={1,2,3} than 2 A and 4 A Set Equality Set A is equal to set B provided the following two conditions are met: 1. Every element in set A is an element is set B 2. Every element in set B is an element in set A Ex: A={1,2,3} B={1,2,1,2,1,2,1,3,3,3,3,3} then A = B 2.2 Venn Diagrams and Subsets Universal set: the set of all elements to be discussed: this is sometimes implied or directly given. Notation: U = universal set Venn diagram: a visual way of depicting the relationship between sets using a rectangle to denote the universal set and circles to denote sets with in the universal set. The Compliment of a Set For any set A within the universal set U, the compliment of A is the set of elements of U that are not elements of A Notation: A’ = compliment of set A = {x | x U and x A} Ex: U = { 1,2,3,4}, A={1,2}, A’ = {3,4} Subset of a Set: Set A is a subset of set B if every element in set a is an element of set B Notation: A B = “Set A is a subset of set B” Ex: A={1,2,3} B={1,2,3,4,5,6} then A B Alternative Def’n for Equality of sets Set A = Set B if A B and B A Proper Subset of a Set: Set A is a proper subset of set B if A B and set A is NOT equal to set B. Notation: A B = “set A is a proper subset of set B” Ex: A={1,2,3} B={1,2,3,4} then A B Finding the number of Subsets and Proper Subsets: The number of subsets of a given set = 2 n if n = # of elements in a set. The number of proper subsets of a given set = 2 n - 1 if n = # of elements in a set. 2.3 Set Operations and Cartesian Products Intersection of Sets: the intersection of sets A and B, written “ A B ”, is the set of all elements common to both set A and set B, or A B = { x | x A and x B } . Ex: If A = {1,2,3} B = {3,4,5} then A B = { 3 } Union of Sets: the union of set A and set B, written “ A B ”, is the set of all elements belonging to either set A or set B or both, or A B = { x | x A or x B }. Ex: If A = {1,2,3} and B = {2,3,4} then A B = {1,2,3,4} Difference of Sets: the difference of set A and set B, written “A – B”, is the set of all elements belonging to set A and not to set B, or A – B = { x | x A and x B } Ex: If A = {1,2,3,} and B = {3,4,5} then A – B = {1,2} Ordered Pairs: in the ordered pair (a,b), a is called the first component and b is called the second component. In general (a,b) (b,a). Two ordered pairs are equal only if there first components are equal and their second components are equal, (a,b) = (c,d) if a = c and b = d. Ordered pairs are NOT sets Cartesian Product of Sets: the Cartesian product of sets A and B, written A x B, is A x B = {(a,b)| a A and b B}. Ex: Let A = {1,2,3} and B = {a,b} then A x B = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)} Cardinal Number of a Cartesian Product: If n(A) = a and n(B) = b, then n(AxB) = n(A) x n(B) = n(B) x n(A) = ab De Morgan’s Laws: for any sets A and B b g 1. A B A B b g 2. A B A B 2.4 Cardinal Numbers and Surveys Cardinal Number Formula: for any sets A and B, n( A B) n( A) n( B) n( A B) n( A B) n( A) n( B) n( A B) Ex: Find n(A B) if n(A) = 3, n(B) = 4, and n(A B) = 2 2.5 Infinite Sets and Their Carndinalities One-to-One Correspondence: there will be a one-to- one correspondence between two sets A and B if each element in set A is paired with an element of set B and Each element in set B is paired with each element in set A. Equivalent Sets: two sets are equivalent, written A~B, if you can put the two sets in a one-to-one correspondence.

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