SETS Lecture by JK Gondwe Aug-11 prob-1 1 OBJECTIVES 1. Define terms 2. Describe a set using standard forms of notation. 3. Learn to use set notation 4. Identify relationships between sets. 5. Perform operations on sets Aug-11 prob-1 2 Basic Set Theory Definitions • A set is a collection of elements • An element is an object contained in a set • Notation: ∈ means “is an element of” ∉ means “is not an element of” • Upper case designates set name • Lower case designates set elements Aug-11 prob-1 3 STANDARD FORMS OF SET NOTATION Sets can be defined in two ways: Roster notation - listing all of the elements. Set-builder notation- defining the rules for membership Aug-11 prob-1 4 Set Theory Roster notation {1, 2, 3, 4, 5, 6, 7, 8,9} Roster notation lists all of the elements (inside curly brackets. Aug-11 Course 2 prob-1 5 Set Theory Set-builder {x | x is a positive even integer < 10} Set-builder notation gives a rule. Read the notation {x|x is a positive integer <10} as “the set of all x such that x is a positive integer <10.” Aug-11 Course 2 prob-1 6 Examples A = {1, 2, 3, 4} 1∈A 2∈A 6∉A z∉A B = {x | x is an even number ≤ 10} 2∈B 4∈B Aug-11 9∉B z∉B prob-1 7 Definitions cont’d • If every element of Set A is also contained in Set B, then Set A is a subset of Set B • A is a proper subset of B if B has more elements than A does Aug-11 prob-1 8 Subsets notation • Note: a subset exists when a set’s members are also contained in another set A B notation: ⊆ means “is a subset of” ⊂ means “is a proper subset of” ⊄ means “is not a subset of” Aug-11 prob-1 9 Universal Set Defn: The set of all elements relevant to a given discussion and is designated by the symbol U Aug-11 prob-1 10 Universal Set- Example The universal set is all COM students, each student is an element in the universal set. Some subsets are: - MBBS, Pharmacy, physiotherapy MLS The universal set is a deck of ordinary playing cards, each card is an element in the universal set. Some subsets are: – face cards, numbered cards, suits Aug-11 prob-1 11 Subset Relationships • A = {x | x is a positive integer ≤ 8} set A contains: 1, 2, 3, 4, 5, 6, 7, 8 • B = {x | x is a positive even integer < 10} set B contains: 2, 4, 6, 8 • C = {2, 4, 6, 8, 10} set C contains: 2, 4, 6, 8, 10 • Subset Relationships (True or False) A⊆A A⊄B A⊄C B⊂A B⊆B B⊂C C⊄A C⊄B C⊆C Aug-11 prob-1 12 Set Equality Two sets are equal if and only if they contain precisely the same elements i.e. if A and B are two sets then A=B if every member of A is a member of B and every member of B is a member of A • The order in which the elements are listed is unimportant. • Repetition of elements is irrelevant. Example: If A = {1, 2, 3, 4} A = B and B = A but Aug-11 B = {1, 4, 2, 3} C = {1, 2, 3} A ≠ C and B ≠ C prob-1 13 Cardinality of Sets • Cardinality refers to the number of elements in a set • A finite set has a countable number of elements • An infinite set has at least as many elements as the set of natural numbers • notation: |A| or n(A) represents the cardinality of Set A Aug-11 prob-1 14 Finite Set Cardinality Set Definition Cardinality A = {x | x is a lower case letter} |A| = 26 B = {2, 3, 4, 5, 6, 7} |B| = 6 C = {x | x is an even number < 10} |C|= 4 D = {x | x is an even number ≤ 10} |D| = 5 Aug-11 prob-1 15 Infinite Set Cardinality Set Definition Cardinality A = {1, 2, 3, …} |A| = ∞ B = {x | x is a point on a line} |B| = ∞ C = {x| x is a point in a plane} |C| = ∞ Aug-11 prob-1 16 The Empty Set • Any set that contains no elements is called the empty set • the empty set is a subset of every set including itself • notation: { } or φ Examples ~ both A and B are empty A = {x | x is a positive number < 0 } B = {x | x is a MBBS MLS student} Aug-11 prob-1 17 Set Theory Notation Symbol Meaning Upper case Lower case { } designates set name designates set elements enclose elements in set ∈ or ∉ ⊆ ⊂ ⊄ | or : | | or n(A) Aug-11 is (or is not) an element of is a subset of (includes equal sets) is a proper subset of is not a subset of such that (if a condition is true) the cardinality of a set prob-1 18 The Power Set ( P ) • The power set is the set of all subsets that can be created from a given set • The cardinality of the power set is 2 to the power of the given set’s cardinality • notation: P (set name) Example: A = {a, b, c} where |A| = 3 P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, φ} and |P (A)| = 8 In general, if |A| = n, then |P (A) | = 2n =23 Aug-11 prob-1 19 Special Sets • Z represents the set of integers – Z+ is the set of positive integers and – Z- is the set of negative integers • N represents the set of natural numbers • ℝ represents the set of real numbers • Q represents the set of rational numbers Aug-11 prob-1 20 Natural Numbers • Counting numbers – Begin with 1 – Each successive number is found by adding 1 to the previous number – {1,2,3,4,5,6,7,…} Aug-11 prob-1 21 Whole Numbers • The set of whole numbers is almost the same as the set of natural numbers, the difference is the addition of the number 0. • Each successive number is found by adding 1 to the previous number. • {0,1,2,3,4,5,…} Aug-11 prob-1 22 Integers • The integers consist of the set of whole numbers and their opposites. • Opposite=Additive inverse – Example: The opposite of 3 is -3 • {…,-4,-3,-2,-1,0,1,2,3,4,…} Aug-11 prob-1 23 Rational Numbers • The set of fractions, repeating decimals, and terminating decimals. • Any number that can be represented as a fraction with an integer numerator and integer denominator. • Cannot use roster notation • Aug-11 prob-1 24 Irrational Numbers • The set of all non-terminating, non-repeating decimals. • Numbers cannot be expressed as fractions with integer numerators and integer denominators. • Examples: – Aug-11 0.13482472454...,0.1212212221..., 2 , π , e prob-1 25 Real Numbers • The set of all natural, whole, rational, and irrational numbers. • Includes all of the sets previously described. • Aug-11 prob-1 26 Real Numbers Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Aug-11 Irrational Numbers Every set is contained in the set above, this means that all of the natural numbers are in the whole numbers, all of the whole numbers are in the integers, all of the integers are in the rational numbers and all of the rational numbers are in the real numbers. Additionally all of the irrational numbers are in the real numbers. prob-1 27 COMPLEMENT OF A SET The rectangle represents the universal set, U, while the portion bounded by the circle represents set A. A U Aug-11 prob-1 28 Cont’d The colored region inside U and outside the circle is labeled A' (read “A prime”). This set, called the complement of A, contains all elements that are contained in U but not in A. A′ A U Aug-11 prob-1 29 Complement of a Set For any set A within the universal set U, the complement of A, written A', is the set of all elements of U that are not elements of A. That is A′ = {x | x ∈ U and x ∉ A}. Aug-11 prob-1 30 INTERSECTION OF SETS Notation: Intersection ( I ) ( and ) The elements which are common to both sets. Aug-11 prob-1 31 INTERSECTION OF SETS -example A = { 1, 2, 3 ,4, 5} B = { 2, 4, 6, 8, 10} C = { 5, 6, 7, 8} 1) A I B ={ 2, 4} 4) (A I B) I C { 2, 4} I { 5, 6, 7, 8} { } or “empty set” 2) B I C ={ 6, 8} 3) A I C ={ 5 } Aug-11 prob-1 or O or “null set” 32 UNION OF SETS Notation: Union ( U ) ( or ) - the combined set of elements from two sets with no duplication of elements. Aug-11 prob-1 33 UNION OF SETS -example Given A = { 1, 2, 3 ,4, 5} B = { 2, 4, 6, 8, 10} C = { 5, 6, 7, 8} 1) A U B = { 1, 2, 3, 4, 5, 6, 8, 10} 2) B U C = { 2, 4, 5, 6, 7, 8, 10} 3) A U C = { 1, 2, 3, 4, 5, 6, 7, 8} Aug-11 prob-1 34 Example 1 Set Operations with Three Sets • Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 3, 4, 5} B = {1, 2, 3, 6, 8} C = {2, 3, 4, 6, 7} • Find A ∩ (B U C’) • Solution Find C’ = {1, 5, 8, 9} • Find (B U C’) = {1, 2, 3, 6, 8} U {1, 5, 8, 9} = {1, 2, 3, 5, 6, 8, 9} • Find A ∩ (B U C’) = {1,2,3,4, 5} ∩{1, 2, 3, 5, 6, 8, 9} = {1, 2, 3, 5} Aug-11 prob-1 35