More Sets Exercises with complete solutions

Transcription

1 More Sets Exercises with complete solutions Introduction 1 Let us define the following sets: M = {males}, = {burglars}, S = {unsuccessful people}, P = {members of the police-force}, H = {happy people}where the universal set is the set of all people Write a full description of the following people a b S c P d H e H f H g HP h M P i S H a = {non-buglars} b S = {successful people} c P = {people who are not in the Police} d H = {unhappy people} e H = {happy burglars} f H = {unhappy burglars} g HP = {happy police} h M P {police women} i S H= {successful, unhappy people} 2 Let us define the following sets: R = x 0 x 6, x = 2k + 1, k Z} S = x x 2-7x + 10 = 0} W = 2, 4, 5 etermine the following sets: a RT b RS c RS d ST e RT f RS g TW R = 1, 3, 5, S = 2, 5, T = 2, 4, W = 2, 4, 5 a RT = 1, 2, 3, 4, 5 b RS = 1, 2, 3, 5 c RS = 5 d ST = 2 e RT = 1, 3, 5 = R f RS = 1, 3 g TW = 3 What can you conclude if the following holds for where are subsets of H?

2 a b c d a = b c d = 4 Which descriptions define a set? a members of your class, b the real roots of an equation, c the greatest known prime number, d those boys in your class who are quite high, e cities in lbania, f equations which have one root equal to 2 ll except for d 5 Which sets are infinite? a Prime numbers b Even prime numbers c The vertices of a given polygon d Points of a line segment e Numbers whose digits add up to 13 a, d 6 Let us define the following sets: = Z, = {x x = 2k, k Z} = {x x = 3k, k Z} Find the elements of a b c d e a = {x x = 6k, k Z }, b = {x x = 2k or inclusive x = 3k, k Z }, c = {m m 3k, k Z}, d = {d d = 3k, d 2k, k Z }, e = {m m 6k, k Z } 7 How many subsets of = {1, 2, 3, 4, 5} have either 2 or 3 or both among their elements? has 2 5 = 32 subsets = {1, 4, 5} has 2 3 = 8 subsets Therefore the answer is 32 8 = 24 8 Let H = {x x Z 0 < x < 100 } How many elements are there in the union of all subsets of H? 99 elements

3 omplement Let,, 9 Show that bsorption rule 10 Show that 11 Show that 12 Show that 13 Show that 14 Show that = by absorption 15 Show that = = = 16 Show that 17 Show that 18 Show that

4 = = 19 Show that 20 Show that 21 Show that 22 Show that 23 Show that = 24 Show that ['']' = '' = ''

5 25 Show that = 26 Show that ' = ' = = ' = ' = = 27 Show that = 28 Show that = 29 Show that Set ifference

6 30 Show that 31 Show that 32 Show that Venn diagram suggests that it is false ounterexample is required to disproop the statement: = 1, 2, 3, 9 = 2, 4, 5, 8 = 1, 2, 5, 7 4, 8 1, 2, 3, 4, 8, 9 1, 2, 3, 4, 5, 8, 9 3, 4,8,9 3, 4, 8, 9 Therefore 33 Show that

7 34 Show that 35 Show that = 36 Show that 37 Show that 38 Show that 39 Show that

8 40 Show that 41 Show that = by absorption 42 Show that 43 Show that 44 Show that

9 45 Show that We apply several times therefore 46 Show that 47 Show that 48 Show that or = ''''

10 = '''''' = '' = '' = '''' = '' = '' 49 Show that = []' by absorption = ' 50 Show that 51 Show that =

11 52 Show that = 53 Show that 54 Show that 55 Show that 56 Show that

12 57 Show that 58 Express the complementary sets of the following sets in terms of,,, ', ' by using union intersection only a b c d a or b c d

13 59 ecide which statements are true for every,, set First draw a Venn diagram to make a conjecture then show using algebra on sets or disprove with a counter example a = b = c = d = e a False Let = {1, 2, 3, 5}, = {2, 4, 5}, = {5} = {2, 4}, = {1, 2, 3, 4, 5}, = {1, 2, 3, 4, 5}, = {1, 2, 3, 4} b True = = c True = = = = = = = = = H = d False = {1, 2, 3, 4, 5}, = {5, 4} = {2, 4, 5} e True 60 Show that 61 Show that = 62 Show that

14 or by absorption 63 Show that or by absorption Symmetric difference 64 Show that 65 Show that or = ' = = 66 re the following statements true? a b c d e f Solutions: a, It is true since b, c, d, e, f, are false ounterexample: = {1, 3} = {3, 4} then = {1, 3, 4}, = {3} = {1, 4} 67 Show that = if only if Step 1: We have to show that if then = We know that = implies = Therefore

15 Step 2: We have to show that if = then Let = If x then x Hence x as well This means that x holds So or sing that E F E F since F F 68 Show that = if only if = Step 1: We have to show that if = then = If = then = Therefore are either disjoint sets or = oth imply = We used that if = E then E are disjoint sets or E = Step 2: We have to show that if = then = If = then = = = 69 Show that a = b = if only if 70 a b We know that = So = if only if a ecide whether the following statement is true If then b Is the converse of the statement true? Yes, it is true its converse as well if only if if only if