2 1 Basic Set Concepts Sets A set is a collection or group of objects called elements that can be clearly defined Capital letters are used to name sets and the elements of the set are enclosed in braces Three methods of representing the elements of a set are a word description where the elements are described in words Set N is the set of natural numbers b roster method where each element is listed separated by commas V x y z N 1 2 3 and c set builder notation would describe the natural numbers as N x x I and x 0 read N is the set of all x such that x is a natural number The symbol is used to indicate that an object is a member of a set and the symbol indicates that an object is not an element of a set Examples x V m V The empty set or null set is the set that contains no elements and is represented by or it is a subset of every set A set is finite if the number of elements of the set can be counted 0 elements 5 elements etc It is infinite if the number of elements continues to infinity like the Natural numbers N 1 2 3 The three dots indicates that the elements continue in the same pattern 1 Use all three types of set notation to show the set of whole numbers W a Show whether 5 is a member of W b Show whether 3 is a member of W 2 Use word notation to describe X 3 6 9 3 Use roster notation to describe the set of even Natural Numbers that are less than 10 4 Use set builder notation to describe odd Natural Numbers greater than 100 5 If E 2 4 6 and O 1 3 5 and F 5 10 15 then 5 E 30 F and 17 385 O The cardinal number of a set A n A is the number of elements in set A Sets are equal if they contain exactly the same elements regardless of order Sets are equivalent if they contain the same number of elements that is n A n B Repeating elements in a set does not add a new element to the set 6 Find the cardinal number of each of the following sets A 6 10 10 14 15 16 B 872 C 9 10 11 15 16 D 7 Given the sets X 0 1 1 2 3 5 5 8 Y 1 5 3 0 8 2 and Z 7 8 9 12 which are equal and which are equivalent 2 2 Venn Diagrams and Subsets The Universal Set is a set that contains all the elements under discussion such as the set of all dogs or all musicians or all real numbers Subsets Set A is a subset of B A B if every element of A is also a member of B Set A is a proper subset of B A B if every element of A is also in B but A B A B indicates that A is not a proper subset of B The null set or empty set is a proper subset of every set except itself The number of subsets of a set with n elements is 2n The number of proper subsets of a set with n elements is 2n 1 If V x y z how many subsets and proper subsets can be found 1 List all possible subsets of V above 2 Show whether N the set of natural numbers is a proper subset of W 3 Show whether the set of negative numbers M are a subset of W 4 Define a universal set for D collies poodles terriers C lions tigers leopards 5 Define a universal set of D and C above plus A monkeys birds alligators 6 Given the set A lions tigers bears and wolves determine whether the following statements are true or false a bears A b bears A c bears A d A e A Complement The complement of set A A is the set of elements that are not in set A but are in the universal set that is A x x U and x A If U 0 1 2 and B 0 1 2 100 the complement of B is shown as B 101 102 103 7 If J all months beginning with J find J 8 If the universal set is all integers and N 1 2 3 find N Venn Diagrams A rectangle is used to represent the universal set Sets within the universal set are depicted usually by circles The region outside a circle but within the rectangle is the complement of that set The combination of any set and its complement gives the universal set Let U all the letters of the alphabet and V the vowels draw a Venn Diagram to represent U V and V U 2 1 Class Practice 1 Write a description of the set 4 8 12 16 2 Use roster notation to describe the set x x N and x lies between 42 and 48 inclusive 3 Use set builder notation to describe the set 15 16 17 4 True or false a 20 2 4 6 100 b 12 x N and x 20 5 Fill in the blank with or x x N and x 1 6 Find the cardinality 3 6 9 21 7 True or false a 0 3 6 9 3 6 9 b 4 8 12 16 20 12 4 20 16 8 4 8 Are the sets equal or equivalent explain a A States of the U S B governors of the U S b A 6 10 2 8 4 B 2 2 4 4 4 6 8 8 10 10 10 10 2 2 1 Let U x 0 x 20 A x x 3 N B x 5 x 10 a Find A b Find B c Draw a Venn Diagram 2 Fill in the blank with or a 3 1 5 4 3 5 b t a b b a t 3 Fill in the blank with or a 3 1 5 4 3 5 b t a b b a t 4 True or false 5 List all subsets A 2 4 6 8 10 6 Calculate the number of subsets x 5 x 10 a 2 4 6 8 10 b 2 4 6 8 10