sets and venn diagramsmathsbooks.net/maths quest 11 methods/by chapter... · 2015. 8. 17. ·...

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Chapter 10 Extension — Sets and Venn diagrams 1 Sets and Venn diagrams A set is a collection of objects and is usually referred to by a capital letter. The objects are the elements — or members — of the set and consist of numbers or any symbolic representation of the given information. The elements of a set are separated by a comma and are enclosed by { }. For example, the set A consisting of the vowels of the English alphabet can be written as A = {a, e, i, o, u}. The order of listing the elements is unimportant so we could also write A = {e, i, u, a, o} or A = {u, e, a, o, i} and so on. To indicate that an element belongs to a set we use the symbol . x A is read as ‘x is an element of set A ’ or ‘x belongs to set A ’. Similarly, x A is used to denote that an element does not belong to set A. For example, if W = {Monday, Tuesday} we can write Monday W and Wednesday W . Finite and infinite sets If all the elements of a set can be listed, the set is finite. An infinite set has elements that cannot all be listed. The elements are expressed descriptively or by listing the set’s first few elements and using dots to represent the remaining members of the set. For example, the set N of all positive numbers can be described as: N = {positive even numbers} or N = {2, 4, 6, 8, . . .} Cardinal number The cardinal number of a set A is the number of elements in A and denoted by n(A). For example, if A = {3, 5, 7, 9} then n(A) = 4 and if B = {letters of the English alphabet} then n(B) = 26. Null set and unit set A set containing no elements is called a null set or empty set and is denoted by φ. If the set consists of only one element, it is termed a unit set. For example, the set A = {positive numbers less than 0} can also be written as A = φ and the set B = {even numbers from 3 to 5 inclusive} is a unit set which can also be described as B = {4}. List the elements of the set A = {odd numbers from 3 to 13 inclusive}. THINK WRITE Which are the odd numbers starting at 3 and finishing at 13? A = {3, 5, 7, 9, 11, 13} 1 WORKED Example What is the cardinal number of the set B = {months of the year ending in ER or beginning with J}? THINK WRITE Which are the required months? January, June, July, September, October, November, December Count how many there are. n(B) = 7 1 2 2 WORKED Example

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Page 1: Sets and Venn diagramsmathsbooks.net/Maths Quest 11 Methods/By Chapter... · 2015. 8. 17. · Venn–Euler diagrams The properties of sets can be illustrated by means of a Venn–Euler

C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 1

Sets and Venn diagramsA set is a collection of objects and is usually referred to by a capital letter. The objectsare the elements — or members — of the set and consist of numbers or any symbolicrepresentation of the given information.

The elements of a set are separated by a comma and are enclosed by { }.For example, the set A consisting of the vowels of the English alphabet can be

written as A = {a, e, i, o, u}. The order of listing the elements is unimportant so wecould also write A = {e, i, u, a, o} or A = {u, e, a, o, i} and so on.

To indicate that an element belongs to a set we use the symbol ∈.x ∈ A is read as ‘x is an element of set A’ or ‘x belongs to set A’.Similarly, x ∉ A is used to denote that an element does not belong to set A.For example, if W = {Monday, Tuesday} we can write Monday ∈ W and

Wednesday ∉ W.

Finite and infinite setsIf all the elements of a set can be listed, the set is finite. An infinite set has elements thatcannot all be listed. The elements are expressed descriptively or by listing the set’s firstfew elements and using dots to represent the remaining members of the set.

For example, the set N of all positive numbers can be described as:N = {positive even numbers} or N = {2, 4, 6, 8, . . .}

Cardinal numberThe cardinal number of a set A is the number of elements in A and denoted by n(A). Forexample, if A = {3, 5, 7, 9} then n(A) = 4 and if B = {letters of the English alphabet}then n(B) = 26.

Null set and unit setA set containing no elements is called a null set or empty set and is denoted by φ.

If the set consists of only one element, it is termed a unit set.For example, the set A = {positive numbers less than 0} can also be written as A = φ

and the set B = {even numbers from 3 to 5 inclusive} is a unit set which can also bedescribed as B = {4}.

List the elements of the set A = {odd numbers from 3 to 13 inclusive}.THINK WRITEWhich are the odd numbers starting at 3 and finishing at 13? A = {3, 5, 7, 9, 11, 13}

1WORKEDExample

What is the cardinal number of the set B = {months of the year ending in ER or beginning with J}?THINK WRITE

Which are the required months? January, June, July, September, October, November, December

Count how many there are. n(B) = 7

1

2

2WORKEDExample

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2 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

Equality and equivalence of setsTwo sets A, B are said to be equal if they contain the same elements. We write A = B.

Two sets A, B are said to be equivalent if they have the same cardinal number.We write A ↔ B.For example, if A = {2, 5}, B = {5, 2}, C = {3, 4} then A = B, A ↔ C and B ↔ C.

Universal setA universal set is a set which contains all elements under consideration.

The symbol ε denotes the universal set.For example, the universal set could be ε = {whole numbers} if we are dealing with

A = {even numbers} or we could have ε = {sports that involve a ball} if we are con-cerned with the sets A = {tennis, cricket, baseball} and B = {soccer, basketball}.

Complement of a setThe complement A′ of a set A is the set of all elements of ε not contained in A. Thismeans that A ∪ A′ = ε.

For example, if ε = {1, 2, 3, 4, 5, 6} and A = {1, 2, 3} then A′ = {4, 5, 6}.

Intersection of setsThe intersection of two sets A, B is the set of elements common to A and B and isdenoted by A ∩ B.

For example, if A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A ∩ B = {3, 4}.

Union of setsThe union of two sets A, B is the set that contains all elements belonging to A or B orto both A and B. The union is denoted by A ∪ B. Note that an element in the union islisted only once.

The number of elements in A ∪ B does not necessarily equal n(A) + n(B). If there areelements common to A and B, these would be counted twice when finding n(A) + n(B).

To rectify this, we subtract the number of common elements once, that is:n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

For example, if A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A ∪ B = {1, 2, 3, 4, 5, 6}and n(A ∪ B) = 4 + 4 − 2 = 6.

Disjoint setsSets A and B are called disjoint sets if A ∩ B = φ; that is, if they have no elements incommon.

The sets A = {1, 2} and B = {3, 4} are disjoint sets.For disjoint sets, n(A ∪ B) = n(A) + n(B).

SubsetsIf every element of set A is an element of set B, then A is said to be a subset of B and isdenoted by A ⊂ B. Alternatively, we can say that set B contains set A and write it asB ⊃ A. As an example, if A = {a, b, c} and B = {a, b, c, d, e} then A ⊂ B and B ⊃ A.

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C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 3

If ε = {two digit numbers between 10 and 20}, A = {numbers whose digits add up to 5} and B = {even numbers}, list the elements of:a A b A′ c B d B′ e A ∪ B f A ∩ B g (A ∩ B).′

THINK WRITE

a The only number between 10 and 20 whose digits add up to 5 is 14.

a A = {14}

b List all elements in ε that are not in A. b A′ = {11, 12, 13, 15, 16, 17, 18, 19}

c Elements in set B are the even numbers between 10 and 20.

c B = {12, 14, 16, 18}

d List all elements in ε that are not in B; that is, the odd numbers between 10 and 20.

d B′ = {11, 13, 15, 17, 19}

e List all elements in A or in B or in both. Each element is listed once only. This is all elements in set B.

e A ∪ B = {12, 14, 16, 18}

f The only number in both A and B is 14. f A ∩ B = {14}

g List all elements in ε that are not inA ∩ B.

g (A ∩ B)′ = {11, 12, 13, 15, 16, 17, 18, 19}

3WORKEDExample

If ε = {letters in the sentence THE ROAD IS FUN}, A = {letters in the word THREADS}, B = {letters in the word ITS}, C = {letters in the word FASTER}, list:a (A ∪ B) ∩ C b (A ∪ B ∪ C)′ c (A ∩ B ∩ C).′

THINK WRITE

a Determine A ∪ B. a A ∪ B = {T, H, R, E, A, D, S, I}Decide which elements are common to (A ∪ B) and C.

(A ∪ B) ∩ C = {T, H, R, E, A, D, S, I} ∩ {F, A, S, T, E, R} = {A, E, S, T, R}

b Determine A ∪ B ∪ C. b A ∪ B ∪ C = {T, H, R, E, A, D, S, I, F}Find which elements of ε do not belong in A ∪ B ∪ C. Letters in ε that are not in A ∪ B ∪ C are U, O and N.

(A ∪ B ∪ C)′ = {U, O, N}

c Note which elements (T and S) are common to sets A, B and C.

c A ∩ B ∩ C = {T, S}

Find which elements of ε are not in A ∩ B ∩ C.

so (A ∩ B ∩ C)′ = {H, E, R, O, A, D, I, F, U, N}

12

12

1

2

4WORKEDExample

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4 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

Venn–Euler diagramsThe properties of sets can be illustrated by means of a Venn–Euler diagram (commonlyreferred to as a Venn diagram). It consists of a rectangle representing the universal set εwith all other sets usually represented by circles within the rectangle. The intersectionof sets is the common area of overlapping circles.

Some of the main properties of sets are illustrated by means of the Venn diagramsgiven below.

Universal set

ε ε

εA ⊃

AA'

Complement A'ε

A

Cardinal number n(A) = 5

* * ***

εA

3 ∈ A, 8 ∈ A', 5 ∉ A,

12 3

58

εA B

A ∩ B

εA B

A ∪ B

εB

A

A B⊃

εA B

(A ∪ B)'

A ∩ B' A ∩ B A' ∩ B

List the elements of:a the universal set ε b A c A′ d Be A ∪ B f A ∩ B g (A ∩ B)′ h A′ ∩ B.THINK WRITE

a All elements in the rectangle are in the universal set, ε.

a ε = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

b All numbers in the circle labelled A are in set A. b A = {1, 2, 3, 4, 5}

c A′ is the region outside the circle and contains the elements 6, 7, 8, 9, 10.

c A′ = {6, 7, 8, 9, 10}

d All numbers in the circle labelled B are in set B. d B = {4, 6, 7, 8}

e All elements in A or B or both are in the union A ∪ B.

e A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}

f 4 is in both set A and set B. f A ∩ B = {4}

g List all elements not in A ∩ B that are in ε. g (A ∩ B)′ = {1, 2, 3, 5, 6, 7, 8, 9, 10}

h Elements not in A are 6, 7, 8, 9, 10. Elements in B are 4, 6, 7, 8.

h A′ ∩ B = {6, 7, 8}

Numbers that are in both lists are in A′ ∩ B.

1

2

5WORKEDExample εA B

9

10

23

1

678

5 4

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C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 5

ε = {letters of the alphabet from a to k}, A = {a, c, f, g} and B = {a, f, k}List the elements in:a A ∩ B b A ∪ B c (A ∪ B)′d Draw a Venn diagram for the sets and show that n(A ∪ B) = n(A) + n(B) − n(A ∩ B).THINK WRITE

a Elements in set A and set B are a and f. a A ∩ B = {a, f}

b Elements in A or B or both are a, c, f, g, k. b A ∪ B = {a, c, f, g, k}

c Elements not in A ∪ B that are in ε are b, d, e, h, i, j.

c (A ∪ B)′ = {b, d, e, h, i, j}

d A rectangle will represent ε. d A ∩ B = {a, f} so we require two overlapping circles containing a and f within the rectangle.A = {a, c, f, g} and A ∩ B = {a, f} so the elements c and g lie inside circle A but are outside the region of the intersection of A and B.

B = {a, f, k} and A ∩ B = {a, f} so the element k lies inside circle B but is outside the region of intersection of the two sets.

Write down n(A ∪ B) and the value of n(A) + n(B) − n(A ∩ B) to show that they are the same.

n(A ∪ B) = 5Also, n(A) + n(B) − n(A ∩ B)

= 4 + 3 − 2= 5

12

3

4Figure 11

εA Bi

j

h

bde

c

g fa k

5

6WORKEDExample

If ε = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 9, 10}, B = {1, 5, 6, 8, 9}, C = {1, 3, 6, 8, 10}, display the four sets as a Venn diagram and show that:n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C).THINK WRITE

Represent ε with a rectangle.There will be three overlapping circles to represent the sets A, B and C.

12

7WORKEDExample

Continued over page

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6 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

THINK WRITE

Decide which elements belong to each region.• A ∩ B = {6, 8, 9}, B ∩ C = {1, 6, 8},

A ∩ C = {6, 8, 10}, A ∩ B ∩ C = {6, 8}• 2 and 4 are elements of A but are not

elements of B or C.• 5 is an element of B but is not an

element of either A or C.• 3 is an element of C but is not an

element of A or B.• 9 is an element of A and B but is not

an element of C.• 1 is an element of B and C but is not

an element of A.• 10 is an element of A and C but is not

an element of B.Use the values from the Venn diagram to write down the value ofn(A ∪ B ∪ C) and n(A) + n(B) + n(C)− n(A ∩ B) − n(A ∩ C) − n(B ∩ C)+ n(A ∩ B ∩ C).

n(A ∪ B ∪ C) = 9n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C)− n(B ∩ C) + n(A ∩ B ∩ C)

= 6 + 5 + 5 − 3 − 3 − 3 + 2= 9

3

Figure 12ε

A

C

B2 59

3

6 810 1

4

4

remember1. { } A set is a collection of objects (or elements).2. x ∈ A denotes that element x belongs to set A.3. x ∉ A denotes that element x does not belong to set A.4. A finite set can be listed; an infinite set cannot be listed.5. A null set (denoted φ) contains no elements.6. A unit set contains one element.7. n(A) The cardinal number of a set is the number of elements it contains.8. ε The universal set is a set containing all elements being considered.9. A′ The complement of set A is all the elements of the universal set

not contained in set A.10. A = B Two sets are equal if they contain the same elements.11. A ↔ B Two sets are equivalent if they have the same cardinal number.12. A ∩ B The intersection of two sets is the set of elements common to both

sets.13. A ∩ B = φ Two sets are disjoint sets if they have no elements in common.14. A ∪ B The union of two sets A, B is the set that contains all elements

belonging to A or B or to both A and B.15. n(A ∪ B) = n(A) + n(B) − n(A ∩ B)16. A ⊂ B Set A is a subset of set B if all the elements of set A are contained

in set B.17. A ⊃ B Set A contains set B if set A contains all the elements of set B.18. Properties of sets can be represented as Venn diagrams.

remember

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C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 7

Sets and Venn diagrams

1 List the elements of each set.a N = {numbers on a standard 6-sided die}b M = {months of the year beginning with the letter M or the letter J}c P = {all prime numbers from 5 to 20 inclusive}d C = {the colours of the rainbow}e L = {the letters common to the two words MAGPIES and PREMIERS}

2 Write down the cardinal number of each set.a A = {1, 3, 5, 7, 9}b B = {α, β, γ, δ, ε}c C = {all odd numbers from 10 to 20 inclusive}d D = {all months of the year that have more than 30 days}e M = {all numbers common to the multiples of 4 and the multiples of 3 which are

from 1 to 50 inclusve}

3

If Q = {number of months beginning with the letter J or whose name contains lessthan 6 letters}, then n(Q) is:

4

If A = {letters of the word TOP} and B = {letters of the word PIT}, then A ∩ B is:

5 Decide in each case if set A is a null set or a unit set.a A = {days of the week beginning with the letter w}b A = {number of vowels in the word SKY}c A = {positive even numbers less than 50 that are divisible by 27}d A = {the smallest natural number}

6 For ε = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 4, 6, 8}, B = {1, 3, 5, 6, 7} and C = {5} statewhether each statement is TRUE or FALSE.

7

If A = {15, 85}, B = {25, 35, 65} and C = {15, 35, 65, 75} then ((A ∪ B)′ ∩ (B ∩ C)′)′ is:

8 A = {all odd numbers from 1 to 10 inclusive}, B = {all multiples of 3 from 1 to 10inclusive}, C = {3, 5, 7, 9} and D = {all prime numbers from 3 to 10 inclusive}.

Decide if each statement is TRUE or FALSE.

A 7 B 4 C 8 D 6 E 5

A {T, O, I, P} B {T, P} C {T, O, P, P, I, T} D {T, T, P, P} E {O, I}

a 2 ∈ A b 8 ∉ B c A′ = {3, 5} d B′ = {2, 4, 8}e A ∩ B = {1} f A = B g A ↔ B h ε ⊂ Ai B ⊂ ε j A ∪ B = ε

A {35, 65} B {15, 25, 75, 85} C {15, 25, 35, 65, 75}D {15, 25, 35, 65, 85} E φ

a A = {1, 3, 5, 7, 9} b B ↔ C c A ≠ Cd A ⊂ D e C ⊃ D

10.1WWORKEDORKEDEExamplexample

1

WWORKEDORKEDEExamplexample

2

mmultiple choiceultiple choice

mmultiple choiceultiple choice

mmultiple choiceultiple choice

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8 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

9 ε = {all positive integers less than 25}, A = {factors of 24}, B = {multiples of 4 from1 to 20 inclusive}, C = {squares of the first four natural numbers}List the elements of:

10 a List the elements of the universal set, ε.b List the elements of:

11 List the elements of:

12 List the elements of:

13 A dice is rolled. Let A be the event ‘an even number’ and let B be the event ‘an oddnumber’.a What is A ∩ B?b Illustrate the information using a Venn diagram.

14 A spinner numbered 1, 2, 3, . . . , 20 is spun. A is the event ‘a multiple of 5’ and B isthe event ‘an odd number’.a List A ∩ B.b List A ∪ B.c List (A ∪ B)′.d Illustrate the information using a Venn diagram.

15 A coin and a die are tossed. Let A be the event ‘The coin lands Heads with a numbergreater than 2 on the die’ and let B be the event ‘The coin lands Heads with an evennumber on the die’.a List A ∩ B.b List (A ∪ B)′.c List A′ ∩ B′.d Represent the information as a Venn diagram.

16 Three coins are tossed. A is the event ‘At least one Head’ and B is the event ‘One Tailor two Tails’.a List A ∩ B.b List (A ∪ B)′.c List A ∪ (A ∩ B).d List (B ∩ (A ∪ B))′.e Illustrate the information using a Venn diagram.

a A b A′ c B d B′e C f C′ g A ∪ B h A ∪ Ci B ∪ C j (B ∪ C)′ k B ∩ C l A ∩ Bm (A ∩ B)′ n (A ∪ C) ∩ B o (A ∩ C) ∪ (B ∪ C)′

i A ii B iii A′ iv A ∪ Bv A ∩ B vi (A ∩ B)′ vii A′ ∩ B viii A ∩ B′

a X b Y c X ∪ Yd Y ′ e X ∩ Y f (X ∩ Y)′g X ∪ Y ′ h (X ∪ Y)′ ∩ ε i (X′ ∩ Y) ∪ (X ∩ Y)

a A b B c A ∪ Bd A ∩ B e A ∩ A′ f A ∪ B′

WWORKEDORKEDEExamplexample

3,4

εA Bi

j

hde

fg

abc

WWORKEDORKEDEExamplexample

5

εX Y

6 2 413

75

Figure 15

εB

A

b de

cgh

ia

WWORKEDORKEDEExamplexample

6

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C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 917 One letter is randomly selected from the set {a, b, c, d, e, f, g, h, i, j, k, l, m, n}. Let

A be the event ‘the letter is a vowel or one of the letters from the word MIDDLE’ andlet B be the event ‘the letter is a consonant’.a List A ∩ B.b List (A ∪ B)′.c List A′ ∩ B′.d List (A ∩ B)′ ∪ (A′ ∩ B′).e Illustrate the information using a Venn diagram.

18 a Show the sets ε = {a, b, c, d} and A = {b, d} as a Venn diagram.b Present the sets ε = {7, 9, 10, 14, 18}, A = {9, 14, 18}, B = {7, 9, 14} as a Venn

diagram.c Present the following three sets as a Venn diagram.

ε = {letters of the alphabet from A to M inclusive}, C = {letters in the wordsBAKED CHIN}, D = {letters in the words MAGIC BEND}

19 Depict the sets ε, E, F as a Venn diagram.ε = {whole numbers from 1 to 100 inclusive that are divisible by 8}E = {multiples of 16 from 1 to 64 inclusive}F = {multiples of 32 from 1 to 100 inclusive}a List the elements of (F ∩ E)′.b List the elements of F ′ ∪ E.c By using your results from a and b decide if 32 ∈ (F ′ ∪ E) ∩ (F ∩ E)′.

20 Decide whether each statement is TRUE or FALSE.(Hint: Draw a Venn diagram to illustrate each situation.)a If A ∩ B = φ then A ∩ B′ = A.b If A ⊂ B then A ∩ (B ∩ A) = B.c If A ⊃ B then (A ∩ B) ∩ A = B.d If A ∩ B ≠ φ then A′ ∪ B′ = (A ∩ B)′.e If A ∩ B = φ then A′ ∪ B′ = ε.f If A ∩ B ≠ φ then (A ∩ B′)′ = (A ∪ B)′.

21 a By taking ε = {whole numbers from 1 to 10 inclusive} create 2 subsets A, B suchthat A ∩ B ≠ φ, A ∪ B = ε and display them as a Venn diagram.

b Use your sets to show that n(A ∪ B) = n(A) + n(B) − n(A ∩ B).c Is the expression in b valid if A ⊂ B?

22 If ε = {natural numbers from 1 to 10 inclusive} andA = {1, 2, 3, 4, 5, 9, 10}, B = {2, 3, 6, 7, 9}, C = {1, 3, 6, 8}, find the value of:

a

b Show the sets ε, A, B, C as a Venn diagram.c By substituting the values obtained for part a, verify that:

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C)+ n(A ∩ B ∩ C)

23 If ε = {natural numbers from 1 to 10 inclusive} suggest three subsets A, B, C such thatA ⊂ B, A ∪ B ∪ C = ε, B ∩ C ≠ φ, A ∩ C = φ, and present them as a Venn diagram.

i n(A) ii n(B) iii n(C) iv n(A ∩ B)v n(A ∩ C) vi n(B ∩ C) vii n(A ∩ B ∩ C) viii n(A ∪ B ∪ C)

WWORKEDORKEDEExamplexample

7

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10 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

Using sets to solve practical problemsWhen dealing with practical problems it is often a good idea to first present the giveninformation as a Venn diagram.

Each of 25 students likes at least one of the two sports, football and tennis. If 10 students like football only and 8 students prefer only tennis, how many students like both sports?THINK WRITE

Draw a Venn diagram showing two overlapping sets.

F = {students who like football}T = {students who like tennis}

Let F be the set of students who like football.Let T be the set of students who like tennis.

Let x be the unknown number of students that like both sports.Show on the Venn diagram the 10 students who like only football and the 8 students who like only tennis.Write the expression for the total number of students and solve the equation.

10 + x + 8 = 25so x = 7.

Therefore, 7 students like both football and tennis.

1

2

3

Figure 16

εF T

10 x 8

4

5

6

8WORKEDExample

The members of a group of 100 people were asked which of two types of peanut butter, Smooth or Crunchy, they buy. If 14 people said they buy both styles, 54 people stated they buy Smooth and 10 said they do not buy either type, determine how many people buy only Crunchy peanut butter.THINK WRITE

On a 2-set Venn diagram, let S be the set of people who use Smooth and let C be the set of people who use Crunchy.

S = {people who use Smooth}C = {people who use Crunchy}

Fourteen people buy both brands. Put this number in the area of intersection of the two sets.If 14 people use both types and 54 people use Smooth, there are 40 people who use only Smooth.

There are 10 people who do not use either brand. Place this number in the area outside the 2 sets.Let x be the number of people who buy only Crunchy.Write the expression for the total number of people and solve the equation.

40 + 14 + x + 10 = 100so x = 36.

Therefore, 36 people buy only Crunchy.

1

2

3

Figure 17

εS C

10

40 14 x

4

56

9WORKEDExample

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C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 11

The cafeteria at Dale Secondary College offers three pie desserts: apple, banana and cherry. Of the 493 students who like at least one type of dessert, 203 like apple, 63 like both apple and banana, 218 like banana, 68 like both banana and cherry, 58 like both apple and cherry and 28 like all three types.

Calculate how many students like only cherry pie.

THINK WRITE

Show A, B and C, the sets representing the students who like apple, banana and cherry pie.28 students like all 3 desserts, so n(A ∩ B ∩ C) = 28.63 − 28 = 35 students like only apple and banana pie.58 − 28 = 30 students like only apple and cherry pie.68 − 28 = 40 students like only banana and cherry pie.

203 − (30 + 28 + 35) = 110 students like only apple pie.218 − (35 + 28 + 40) = 115 students like only banana pie.Let x be the number of people who buy only cherry pie.There are no students who have no dessert, so (A ∪ B ∪ C)′ = φ.Write the expression for the total number of students and solve the equation.

x + 110 + 115 + 30 + 35 + 40 + 28 = 493so x = 135

Therefore, 135 students like only cherry pie.

1

2

3

4

5

Figure 18

εA

C

B115110 35

2830 40

x

6

7

8

9

10

10WORKEDExample

rememberTo solve a problem using sets:1. Use overlapping regions (for example, circles) to represent sets within a

universal set (for example, a rectangle).2. Label each set.3. Fill in any given information.4. Calculate required missing quantities.

remember

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12 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

Using sets to solve practical problems

1 Thirty children were asked about eating apples and pears. Fifteen children said theylike to eat only apples while 13 children said they eat only pears. No children saidthey eat any other kind of fruit.a Show the information as a Venn diagram.b How many children eat both apples and pears?

2 A survey of 500 dairy and sheep farmers revealed that 290 farms have sheep only and125 farms have cows only.a Show the information as a Venn diagram.b How many farms have both sheep and cows?

3 An inspection of 50 cars revealed the following information:• Twenty-three cars have airconditioning.• Fourteen cars have airconditioning and a CD player.• Five cars do not have airconditioning or a CD player.a Use the information to draw a Venn diagram.b How many cars have only a CD player?c How many cars have only airconditioning?

4 Two hundred car owners were asked which of two brands of tyre — SureGrip andTitan — they would use. Fifty-five car owners said they would purchase onlySureGrip tyres, 28 would buy both types, while 38 owners stated they would use adifferent brand altogether.

Draw a Venn diagram and calculate the number of car owners who would buy onlythe Titan brand.

5

Of a group of 53 children, each of whom plays at least 1 of 2 musical instruments, 21play the piano and 38 play the guitar. The number of students who play both instru-ments is:

6

A total of 26 athletes are to compete in running events. A certain number of them willtake part in the 100-metre race and twice as many will be in the 400-metre event.Three athletes will not compete in either of these two events. If 8 athletes will run inboth the 100-metre and the 400-metre races, the number who will take part in the 100-metre race is:

7

In a group of 60 students, 20 study Mathematics only. Nineteen students do not studyMathematics or English. There are twice as many students studying only English asthere are students studying both English and Mathematics. The number of studentsstudying English only is:

A 17 B 12 C 7 D 4 E 6

A 5 B 7 C 2 D 8 E 6

A 15 B 13 C 9 D 7 E 14

10.2WWORKEDORKEDEExamplexample

8

WWORKEDORKEDEExamplexample

9

mmultiple choiceultiple choice

mmultiple choiceultiple choice

mmultiple choiceultiple choice

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C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 138 Of 34 families it was found that 15 have only cereal for breakfast and 10 eat only

toast. Six families do not have either cereal or toast.Draw a Venn diagram and find how many families eat both types of food for

breakfast.

9 Ninety students travel to school by bus, train or both. Of these, 63 travel by bus onlywhile some of them travel by both bus and train. Twice as many students travel bytrain only as use both methods of transport.a Display the information as a Venn diagram.b How many students use both a bus and a train to get to school?c How many students travel to school by train only?

10 A survey of 25 children revealed that with their lunch 13 of them drink water and 18drink orange juice. No other type of refreshment is available and each child has atleast one type of drink. How many children drink:a water and orange juice?b only water?c only orange juice?

11 At the end of a debate, 20 members of a 50-member debating club voted for the Yesteam while 22 favoured the No team. There were 8 members in the two debatingteams and they did not vote. How many voted twice, assuming that all members notdebating cast at least one vote?

12 A group of people were asked which type of car they prefer.The Venn diagram shown contains some of the informationobtained.a If 90 people stated they like Holden, how many people

said they like only Ford and Holden?b If 75 people like Toyota, how many said they like Ford

and Toyota only?c If 70 people stated they like Ford and 200 people were interviewed, how many

said they do not like any of the 3 cars?[Hint: You may wish to use n(A ∪ B ∪ C ) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C).]

13 A school offers three sports: football, cricket and soccer. In a group of 30 students:• 6 play all 3 sports• 7 play football and soccer• 5 play soccer only• no student plays cricket and soccer but not football• 12 students play football• 2 play football and cricket only.How many students play only cricket if there are 9 students who play no sport?

14 Forty patrons at a cinema complex wish to buy tickets. Movies screening include AllIs Nothing, I’m Me and Carrying On. One avid fan will see all three movies, 13 willsee All Is Nothing and 11 will watch I’m Me. Additionally, 2 people will view only AllIs Nothing and I’m Me, 6 will watch only All Is Nothing and Carrying On while 5theatregoers are there to see only I’m Me and Carrying On. If 12 people are buyingtickets for Rocky XV, how many patrons will be watching only Carrying On?

Figure 19

ε

Ford

Toyota

Holden35

2914

11

WWORKEDORKEDEExamplexample

10

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14 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

15 Of 347 holiday-makers who visited at least one South American country, 147 visitedArgentina, 100 saw Brazil, 42 saw both Argentina and Brazil, 50 toured Chile andBrazil, 45 travelled through Argentina and Chile and 42 people enjoyed all 3 countries.a Calculate how many travellers went to Chile.b Represent the information as a Venn diagram.

16 Music enthusiasts were asked their preference for three music styles: classical, pop,and rock and roll. All stated that they like at least one of these music styles. From thegroup, 250 like classical, 200 like rock and roll, 210 want pop music, 125 likeclassical and rock and roll, 130 enjoy both classical and pop music and 140 peopleprefer rock and roll and pop music.a If 80 people said they like all three styles, how many were interviewed?b Represent the information as a Venn diagram.

17 A survey of 70 students found that 34 like Maths, 31 like English, 37 enjoy Science,16 are happy doing both Maths and English, 19 students enjoy studying both Mathsand Science and 17 like English and Science. However, 9 students indicated they donot like any of the subjects.a How many students like all three subjects?b Show the information as a Venn diagram.

Page 15: Sets and Venn diagramsmathsbooks.net/Maths Quest 11 Methods/By Chapter... · 2015. 8. 17. · Venn–Euler diagrams The properties of sets can be illustrated by means of a Venn–Euler

C h a p t e r 1 0 E x t e n s i o n — S a m p l i n g w i t h o u t r e p l a c e m e n t 15

Sampling without replacementNote: This section is not specified in the VCE study design. It is included here becauseit relates well with previous sections and is useful as enrichment work.

Miguel randomly takes a marble from a box containing 3 redand 2 blue marbles as shown.

He replaces the marble and withdraws another one.What is the probability that Miguel will take out a red marble

each time?Since the probability of a red marble is each time a marble is withdrawn, and the

two trials are independent, the probability of getting 2 red marbles is × = .Sandra repeats Miguel’s experiment but does not replace the first marble after it has

been withdrawn. What is the probability that Sandra takes out 2 red marbles?In Sandra’s case the number of possible outcomes is reduced after the first marble is

taken out. Therefore there is a different probability of a red marble being chosen thesecond time.

For the first trial, Sandra’s chance of getting a red marble is . We require that on hersecond try there are 2 red marbles left to choose from and 4 marbles in the box. Herprobability of success is therefore × = .

Notice that this calculation is just an application of conditional probability.Recall that:

Pr(A ∩ B) = Pr(A) × Pr(B | A)or in this case,

Pr(red on first trial and red on second trial)= Pr(red on first trial) × Pr(red, given red on first trial [and so 1 less red in bag.])

As stated previously, probabilities may be multiplied for a chain of events, as long as the effect of previous events is taken into account at each stage — that is, always consider reduced event spaces.

35---

35--- 3

5--- 9

25------

35---

35--- 2

4--- 3

10------

Inside a box are 5 red marbles and 3 blue marbles. One marble is randomly taken out, its colour noted and a second marble randomly taken out. Calculate the probability of getting:a a red marble followed by a blue marbleb a red marble and a blue marble (in any order)c 2 red marbles or 2 blue marbles.

THINK WRITE

a Consider the chain of events required and express this in terms of probability. We need a red marble from 8 marbles followed by a blue from the 7 remaining. Substitute values and calculate.

a Pr(red then blue) = ×

=

158--- 3

7---

1556------

1WORKEDExample

Continued over page

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16 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

THINK WRITEb Consider all ways of getting a red and

a blue marble in any order. This can be achieved in two ways: red then blue or blue then red.

b

Express the above in terms of probabilities.

Pr(red and blue) = Pr(red then blue or blue then red)

= Pr(red then blue) + Pr(blue then red)

Substitute values and calculate. = × + ×

= +

=

=

c Write an expression for the required probability.

c Pr(two red or two blue)= Pr(red then red or blue then blue)

Substitute values and calculate. = Pr(red then red) + Pr(blue then blue)

= × + ×

= +

=

=

1

2

358--- 3

7--- 3

8--- 5

7---

1556------ 15

56------

3056------

1528------

1

258--- 4

7--- 3

8--- 2

7---

2056------ 6

56------

2656------

1328------

A bag contains 10 tags numbered 1, 2, 2, 4, 5, 5, 7, 8, 9, 9. Three tags are randomly taken out one at a time.a Find the probability of getting 3 tags each having an even number.b Find the probability of getting the first 2 tags labelled with two even numbers and the

third tag labelled with an odd number.c Find the probability of getting at least 2 tags labelled with an odd number.d Find the probability of selecting no even numbers. THINK WRITEa Write an expression for the required

probability.a Pr(3 even numbers)

Consider the chain of events and probabilities with corresponding reduced event spaces. There are initially 4 even and 6 odd numbers in the group of 10.

= × ×

Cancel where possible and simplify. =

1

2410------ 3

9--- 2

8---

3 130------

2WORKEDExample

Page 17: Sets and Venn diagramsmathsbooks.net/Maths Quest 11 Methods/By Chapter... · 2015. 8. 17. · Venn–Euler diagrams The properties of sets can be illustrated by means of a Venn–Euler

C h a p t e r 1 0 E x t e n s i o n — S a m p l i n g w i t h o u t r e p l a c e m e n t 17 THINK WRITE

b Write an expression for the required probability.

b Pr(even, even, odd)

Consider the chain of events and probabilities with corresponding reduced event spaces.

= × ×

Cancel where possible and simplify. =

c Write an expression for the required probability.

c Pr(at least 2 odd)= Pr[(odd, odd, even) or (odd, even, odd) or

(even, odd, odd) or (odd, odd, odd)]= Pr(odd, odd, even) + Pr(odd, even, odd)

+ Pr(even, odd, odd) + Pr(odd, odd, odd)

Consider the chain of events and probabilities with corresponding reduced event spaces.

=

+

Cancel where possible and simplify. =

d Write an expression for the required probability.

d Pr(no even) = Pr(3 odd)

Consider the chain of events and probabilities with corresponding reduced event spaces.

= × ×

Cancel where possible and simplify. =

1

2410------ 3

9--- 6

8---

3110------

1

2610------ 5

9--- 4

8---××( ) 6

10------ 4

9--- 5

8---××( ) 4

10------ 6

9--- 5

8---××( )+ +

610------ 5

9--- 4

8---××( )

3 23---

1

2610------ 5

9--- 4

8---

3 16---

remember1. Sampling without replacement involves objects selected one at a time.2. An object that is selected is no longer part of the event space, so that the event

space is reduced.3. The probability of a chain of events is obtained by multiplying the individual

probabilities, taking into account the reduced event space at each stage.

remember

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18 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

Sampling without replacement

1 A bag contains 10 beads numbered 1 to 10. Two beads are randomly withdrawn oneat a time. Calculate the probability of getting:a a bead numbered 6 followed by a bead numbered 7b two even-numbered beadsc a bead which is a multiple of 3 followed by a bead which is a multiple of 5.

2

A box contains 20 balls numbered 1, 2, 3, . . . , 20. Two balls are taken out at random,one at a time. The probability that both numbers on the balls are even is:

3 In a box are 4 red discs, 2 blue discs, 3 white discs and 1 green disc. Three discs arerandomly taken out. Find the probability of getting:a 3 white discsb 2 red discs followed by 1 blue discc 2 white discs followed by 1 green disc or 2 blue discs followed by 1 red discd no red discs.

4 From a pack of 52 playing cards, 2 cards are drawn. Find the probability that:a both cards are blackb the first card is a spade and the other is the king of heartsc both cards are picture cards or both cards are 4.Note: A picture card is one of jack, queen, king.

5

A captain and a vice-captain are to be selected from 6 girls and 5 boys. The prob-ability that both positions will be filled by 2 boys or 2 girls is:

6 Inside Adam’s pocket are 2 dice, 3 lollies and 5 coins. Three of the objects fall out. Ifeach object is equally likely to fall out, what is the probability that:a the objects are 3 coins?b the first two objects are dice and the third object is a coin?c at least one of the objects is a coin?d none of the three objects are coins?

7 A jewellery box contains 4 necklaces, 7 rings, 6 bracelets and 3 watches. Two itemsare randomly selected. Determine the probability that:a both items are necklacesb one of the items is a bracelet and the other is a watchc both objects are rings or one item is a watchd at least one of the items is a bracelete only bracelets and watches are selected.

A B C D E

A B C D E

10.3 SkillSH

EET10.2

WWORKEDORKEDEExamplexample

1

mmultiple choiceultiple choice

14--- 9

20------ 9

38------ 1

9--- 7

19------

WWORKEDORKEDEExamplexample

2

mmultiple choiceultiple choice

911------ 2

9--- 28

55------ 1

3--- 5

11------

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C h a p t e r 1 0 E x t e n s i o n — S a m p l i n g w i t h o u t r e p l a c e m e n t 198 A 3-member school committee is to be selected from a group consisting of 6 students,

4 teachers and 2 parents. If each member is randomly selected what is the probabilityof forming a committee comprising:a 3 students?b at least 2 teachers?c 2 parents or at least 2 students?d no students?

9

Derryn, Jenny and Carmen each have a lottery ticket. A first, second and third prizeare to be won. If their names have been placed in the final draw consisting of 20people in total, the probability that Derryn, Jenny and Carmen receive at least oneprize between them is:

10 On Aaron’s bookshelf are 3 Maths textbooks, 4 English novels, 6 fiction paperbacksand 2 dictionaries.

If Aaron randomly selected 4 books to peruse, what is the probability that heselected:a 2 fiction paperbacks followed by 2 Maths texts?b at least 3 English novels given that 2 English books were the first books chosen?

11

Two jelly beans are taken out of a bag containing 5 red, 2 green and 3 yellow jellybeans without replacement. The probability of getting at least one yellow jelly bean is:

12

If two marbles are randomly taken out one at a time without replacement from a jarcontaining 3 red, 5 blue and 2 white marbles, the probability of getting one blue andone white marble is:

A B C D E

A B C D E

A B C D E

mmultiple choiceultiple choice

16--- 23

57------ 3

4--- 17

20------ 1

5---

mmultiple choiceultiple choice

15--- 3

30------ 3

5--- 2

3--- 8

15------

mmultiple choiceultiple choice

845------ 7

45------ 7

90------ 4

9--- 11

90------

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20 A n s w e r san

swer

sCHAPTER 10 ProbabilityExercise 10.1 — Sets and Venn diagrams1 a N = {1, 2, 3, 4, 5, 6}

b M = {January, March, May, June, July}c P = {5, 7, 11, 13, 17, 19}d C = {red, orange, yellow, green, blue, indigo,

violet}e L = {M, P, I, E, S}

2

3 D4 B5

6

7 D8

9 a {1, 2, 3, 4, 6, 8, 12, 24}b {5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21,

22, 23}c {4, 8, 12, 16, 20}d {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19,

21, 22, 23, 24}e {1, 4, 9, 16}f {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19,

20, 21, 22, 23, 24}g {1, 2, 3, 4, 6, 8, 12, 16, 20, 24}h {1, 2, 3, 4, 6, 8, 9, 12, 16, 24}i {1, 4, 8, 9, 12, 16, 20}j {2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 18, 19, 21,

22, 23, 24}k {4, 16}l {4, 8, 12}m {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18,

19, 20, 21, 22, 23, 24}n {4, 8, 12, 16}o {1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 14, 15, 17, 18, 19,

21, 22, 23, 24}10 a ε = {a, b, c, d, e, f, g, h, i, j}

b

11

12

13 a A ∩ B = Φb

14 a {5, 15}b {1, 3, 5, 7, 9, 10, 11, 13, 15, 17, 19, 20}c {2, 4, 6, 8, 12, 14, 16, 18}d

15 a {H4, H6}, H is Headb {H1, T1, T2, T3, T4, T5, T6}, T is Tailc {H1, T1, T2, T3, T4, T5, T6}d

16 a {HHT, HTH, THH, TTH, THT, HTT}, H is Head, T is Tail

b {TTT}c {HHH, HHT, HTH, THH, TTH, THT, HTT}d {HHH, TTT}e

17 a {d, l, m} b φ c φd {a, b, c, e, f, g, h, i, j, k, n}e

18 a b

c

19 a {8, 16, 24, 40, 48, 56, 72, 80, 88, 96}b {8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88}c 32 ∉ (F ′ ∪ E) ∩ (F ∩ E)′

l n(A) = 5 m n(B) = 5 n n(C) = 5o n(D) = 7 p n(M) = 4

a Unit set b Null setc Null set d Unit seta True b True c Falsed True e False f Falseg True h False i Truej True

a True b False c Trued False e True

i {a, b, c, d, e} ii {a, b, c, f, g}iii {f, g, h, i, j} iv {a, b, c, d, e, f, g}v {a, b, c} vi {d, e, f, g, h, i, j}

vii {f, g} viii {d, e}a {1, 2, 3, 4, 7} b {4, 5, 7}c {1, 2, 3, 4, 5, 7} d {1, 2, 3, 6}e {4, 7} f {1, 2, 3, 5, 6}g {1, 2, 3, 4, 6, 7} h {6}i {4, 5, 7}

a {a, b} b {a, b, c, d, e}c {a, b, c, d, e} d {a, b}e φ f {a, b, i, g, h}

Figure 121

εBA

2 46

1 35

Figure 122

εA B

1 3 79 1113 17

19

515

1020

8 12 14 16 18

24 6

Figure 123

εA B

H3H5

H4H6

H2H1T1 T2 T3 T4T5 T6

Figure 124

εA B

HHHHHTHTHTHH

HHTHTHTHH

TTHTHTHTT

TTT ε A

B

HHTHTH THHTTH THTHTT HHTHTH THH

TTTHHH

Figure 125

εA B

b cf g hj k n

a e ilmd

Figure 89

εA

d bac

Figure 90

εA B

18 79 10

14

Figure 91

εC D

KH

MG

BADECIN F

J L

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A n s w e r s 21

answ

ers

20

21 a As an example take A = {1, 2, 3, 4, 5}, B = {1, 2, 3, 6, 7, 8, 9, 10}.

b n(A) = 5, n(B) = 8, n(A ∩ B) = 3n(A ∪ B) = 10 and n(A) + n(B) − n(A ∩ B) = 5 + 8 − 3 = 10

c Yes.22 a

b

c n(A ∪ B ∪ C) = 10n(A) + n(B) + n(C) − n(A ∩ B)− n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C) = 7 + 5 + 4 − 3 − 2 − 2 + 1 = 10

23 One example is A = {1, 2}, B = {1, 2, 3, 4, 5, 6} and C = {5, 6, 7, 8, 9, 10}.

Exercise 10.2 — Using sets to solve practical problems1 a b 2

2 a b85

3 a b 22 c 9

4 79

5 E6 A

7 E8 3

9 a b 9 c 18

1011 None

1213 Four students play only cricket.14 Seven patrons will be watching Carrying On.15 a 195 people visited Chile.

b

16 a 345 people were interviewed.b

17 a 11 students like all three subjects.b

Exercise 10.3 — Sampling without replacement

1

2 C

3

4

5 E

6

7

8

9 B10

11 E 12 D

a True b False c Trued True e True f False

i 7 ii 5 iii 4 iv 3v 2 vi 2 vii 1 viii 10

εA B

45

6 78 910

123

Figure 93

εA

C

B

81 63

72 94 10

5

εB C

1 2A 5

67 89103 4

Figure 95

ε

Apple Pear290 x 13

Figure 96

εS C

290 125x

Figure 97

εAir

conditioning CD

9x

5

14(23)

Figure 98

ε

SureGr Titan

55 x

38

28

a 6 b 7 c 12

a 12 b 21 c 55

a b c

a b c d

a b c

a b c d

a b c d e

a b c d

a b

Figure 99

ε

Cereal Toast

15 10

6

x

Figure 100

εBus Train

63 2xx

Figure 101

εArgent.

Chile

Brazil50

0

142

423 8

102

Figure 102

εClassic

Rock

Pop2050

15

8045 60

75

Figure 103

εMaths English

95

12

118 6

10

Science

190------ 2

9--- 1

15------

1120--------- 1

30------ 7

360--------- 1

6---

25102--------- 1

204--------- 12

221---------

112------ 1

72------ 11

12------ 1

12------

395------ 9

95------ 36

95------ 99

190--------- 18

95------

111------ 13

55------ 6

11------ 1

11------

1182--------- 23

78------