sets.pdf

7/27/2019 sets.pdf

http://slidepdf.com/reader/full/setspdf 1/20

Chapter 1 – Set Language and Notation 1

Chapter 1 – Set Language and Notation

1. Definitions and Notations 2. Venn Diagrams 3. Problem Solving

P Logic

Definitions and Notations 1-1

1. A set is a collection of well-defined (distinct) elements. Each letter or number in the set iscalled the element of the set.

2. We use capital letters to represent sets. We list its elements by enclosing them within bracesseparated by commas.

3. If the number of elements of a set is finite, we can list them.

4. If the number of elements of a set is infinite, we can describe the elements.

5. Here is a table of notations used in the language of sets:

Notation What it represents What it means

φ or {} Null set or empty set A set with no elements

ε Universal set A set which consists of all elements

B A∪ Union of two sets All elements in A and B

B A∩ Intersection of two sets Elements in both A and B

n(A) Number of elements in a set Number of elements in set A

∈ “…is an element of…” ∈ A x is in the set A

∉ “…is not an element of…” ∉ A x is not in the set A

A’ Complement of set A All the other elements in the universal setthat are not in the set A

B A⊆ Subset A is a subset of B

B⊂ Proper subset A is a proper subset of B

B A⊄ Not a proper subset A is not a proper subset of B

A = B Equal Sets A and B have exactly the same elements.

6. Useful notations.7.

Notation What it represents

ℜ Is the set of real numbers

+ℜ Is the set of positive real numbers

7/27/2019 sets.pdf


Chapter 1 – Set Language and Notation2

N Is the set of natural numbers { 1 , 2 , 3 , ….}

Z Is the set of integers {…, -3 , -2 , -1 , 0 , 1 , 2, 3 … }

+ Z Is the set of positive integers { 1 , 2 , 3 , ….}

8.

9.

Example 1-2

A = {a, c, f, g } B = {1, 4, 8, 10, 15 }

There are 4 elements in set A and 5 elements in B, thus we write n(A) = 4 and n(B) = 5.

” f ” is an element of set A, we write f ∈ A.“ b ” is not an element of set A, we write b∉ A.

We can also write sets in the following manner.

A = { : is a prime number and < x 20}

B = { : is integer and 123 <<− x }

Universal set

Universal set is a set which consists of all elements represented by ε

Complement set

Other elements in the universal set but not in A is known as the complement of A or A’..

Equal Sets, Subsets and Proper Subsets

Two sets are equal if they have exactly the same elements.

Example 1-3

ε = { : is an integer and 41 ≤≤ x }

Therefore the elements of the universal set is ε = { 1 , 2 , 3 , 4 }

Given A = { 1 , 2 , 3 }, B = { 1 , 2 , 3 }, C = { 2 , 3 }, D = { 3 , 4 }

Then A = B, A ≠ C, A ≠ D, etc

In the above example, C ⊂ A since all the elements in C are also in A.

What is the difference between a subset “⊆” and a proper subset “⊂” ?

A ⊆ B means that all the elements in A are in B, and A may be equal to B

A ⊂ B means that all the elements in A are in B, but A is not equal to B

So, in the above example, the statement A ⊆ B is true, but A ⊂ B is false. C ⊆ A is true, and C ⊂ A isalso true.

7/27/2019 sets.pdf



Example 1-4

ε = { x x : is an integer 101 ≤≤ x }

A = { 72: > x x }

We list down the elements of A and it’s complement.

A = { 4 , 5 , 6 , 7 , 8 , 9 , 10 } A’ = { 1 , 2 , 3 }

Practice 1 1-5

1. Write the following into sets.

a) M denotes the set of months in a year with 30 daysb) G denotes the set of prime numbers from 1 to 50c) H denotes the set of continents on Earth.

2. List the elements in the following set notation.

a) A = {+

Ζ∈ x x :6 and 134 <<− x }

b) B = { : is an even positive integer and 5< x }

c) C = { Ζ∈<<− x x x ,63: }

3. ε = {whole numbers from 1 to 20 inclusive} A = {all even numbers} B = {all odd numbers}C = {all multiples of 5} D = {all multiples of 10}E = {all multiples of 25}

State whether the following are True (T) or False (F):

(a) ε = {1, 2, 3, 4, 5, 6,………., 18, 19, 20} ( )

(b) A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} ( )

(c) B = {1, 3, 5} ( )

(d) 5 ∈ B ( )

(e) 5 ∉ C ( )

(f) n(A) = 10 ( )

(g) n(C) = 8 ( )

(h) D = Ø ( )

(i) E = { } ( )

(j) C ≠ Ø ( )

(k) 5 ∉ D ( )

(l) 5 ∉ Ø ( )

7/27/2019 sets.pdf



4. ε = {whole numbers from 1 to 20 inclusive} A = {all even numbers} B = {all odd numbers}C = {all multiples of 5} D = {all multiples of 10}E = {all multiples of 25}

State whether the following are True (T) or False (F):

(a) A and B are disjoint ( )(b) C and D are disjoint ( )

(c) C ⊆ B ( )

(d) A ⊆ B ( )

(e) C ⊆ D ( )

(f) D ⊆ C ( )

(g) A ⊄ C ( )

(h) D ⊂ C ( )

(i) A' = B ( )

(j) A' ⊂ B ( )

(k) A' ⊆ B ( )

Venn Diagrams 1-6

From example 3, we can represent the set using a Venn Diagram.

ε = { : is an integer 10≤≤1 x }

A = { 4, 5, 6, 7, 8, 9, 10 } A’ = { 1, 2, 3 }

In Venn diagrams, the universal set is always represented by a rectangle.

A ⊆ B is represented here:

Two sets are disjoint = A' = complement of A =they have no common elements set of elements NOT in A

A

B

A

B

A A’

4 5 6

7 8 9

10

A

A’

1 2 3

7/27/2019 sets.pdf



Union and Intersection

A ∩ B = intersection of A and B A ∪ B = union of A and B= set of elements in both A and B = set of elements in A or B

A B

Example 1-7

A = { those who eat apple} B = { those who eat banana}

A ∪ B = { those who eat apple or banana or both} A ∩ B = { those who eat both apple and banana} A ∩ B’ = { those who eat apple and do not eat banana}

A ∩ B ≠ ∅ ⇒ { those who eat apple and banana}is not an empty set

⇒ there are people who eat both apple and banana

A ∩ B’ = ∅ ⇒ { those who eat apple and not banana}is an empty set

⇒ nobody eat only apple and not banana

A ⊆ B ⇒ those who eat apple is contained inside the set of those who eat banana

⇒ all those who eat apple also eat banana.

n(A∪ B) = 14 ⇒ there are 14 people who eat either apple or banana or both.

Practice 2 1-8

1. List the elements of the following sets.

(a) { : is a whole number and 3 x + 2 = 14}

(b) { : is a positive integer and 2 + 1 ≤ 16}

(c) { : is a natural number, = 3 and 2 ≤ y ≤ 10}

2. If A = { months of the year (2006)}, B = { months with only 30 days}, C = {months with less than30 days}, D = {months with 31 days} and E = {months with less than 31 days}

(a) List the elements of A, B, C, D and E

(b) State whether the following statements are true or false.

(i) B ⊆ E ( ) (ii) C ⊂ E ( )(iii) E ⊂ A ( ) (iv) D ⊆ B ( )

(v) D ⊆ A ( ) (vi) C ⊂ D ( )

(vii) B ⊆ D ( ) (viii) D ⊄ A ( )

3. If A = { 1, 3, 5, 6} , B = { 2, 4, 6, 8} and C = {1, 2, 3, 4}, list the elements of the following:

(a) A ∩ B (b) A ∪ B

(c) A ∩ C (d) B ∪ C

(e) A ∩ B ∩ C (f) A ∪ B ∪ C

A B

7/27/2019 sets.pdf



4. ε = { Sec 3 students in ABC school }, A = {students who passed their A Maths test}, B ={students who passed their Biology test }, C = {students who passed their Chemistry test}

(i) Describe the following sets in words:

(a) A ∩ B(b) C’

(ii) Explain the following statements in words:(a) A ∩ B = Ø

(b) A ∪ C = ε

(c) C’ ⊆ A

(d) B ⊆ C

(e) n(A) = n(ε)

(iii) Express the following in set notation:(a) All students passed the Chemistry test(b) Nobody failed both A Maths and Chemistry(c) All students who passed Chemistry also passed A Maths(d) Nobody passed all 3 tests(e) All students passed at least 1 test(f) 120 students passed both A Maths and Biology

(g) 15 students failed Chemistry and Biology(h) 32 students passed Chemistry but failed Biology

Shading Venn Diagrams 1-9

A ∪ B’ A ∩ B’ (A ∩ B’) ∪ (A’ ∩ B)(Either in A or outside B) (in A and outside B)

(A ∪ B)’ or A’ ∩ B’ A∩ B∩C (A∩ B∩C’) ∪ (C∩ B∩ A’)

(A∩B) ∪ C’

A B A B

A B

C

A B

A B

A B

A B

C

7/27/2019 sets.pdf



Practice 3 1-10

1 Shade the regions as indicated:

(a) A ∩ B'

(b) A ∪ B'

(c) A ∩ B ∩ C

(d) (A ∪ B) ∩ C

A B BA

A B BA

BA

C

A B

C

A B

C

BA

C

7/27/2019 sets.pdf



2. Give the conditions in set notation that result in the following Venn Diagrams:

3. Label the shaded region using set notations.(a) (b)

(c) (d)

4. A = { months of the year}, B = { months with only 30 days}, C = {months with less than 30 days},D = {months with 31 days}, E = {months with less than 31 days}Draw a Venn Diagram to illustrate the relationship between the above sets. Write the number of elements in each section of the Venn Diagram.

5. ε = { all triangles} , I = { isosceles triangles},E = {equilateral triangles}, R = {right-angled triangles}, A = {acute-angled triangles}

Draw a Venn Diagram to illustrate the relationship between the above sets.

A B

C

A B

C

A B

C

A B

C

A B C A B C

A B CA B

C

(a) (b)

(c) (d)

7/27/2019 sets.pdf



Problem Solving -- Examples 1-11

Example1 :

1. If n (A ∪ B) = 44, n (A∩B) = 5 and n(A) = 20. Draw a Venn Diagram and find the value of n(B).

Solution:

A 5 B20 – 5 = 15 44 – 15 – 5

=24

n(B) = 5 + 24 = 29

Example 2:

In a class of 40 students, 30 passed their Maths test and 35 passed their Science test.(a) Find the maximum and minimum number of students who passed both tests(b) Find the maximum and minimum number of students who failed both test(c) Find the maximum and minimum number of students who passed Maths but failed Science(d) If 2 students failed both test, how many students passed both tests?

Solution:

Let ε = {all 40 students in the class} and M = {students who passed Maths test} and S = {studentswho passed Science test}.Let x be the number of students who passed both tests.Then,

30 – x x 35 – x

M S

x – 25

ε

Number who failed both = 40 – (30– x ) – x – (35– x ) = x – 25

(a) To ensure that none of (30– x ), x , ( 35– x ) and ( x – 25) become a negative number,Max x = 30 and min x = 25Therefore, max. number to pass both = 30 and min number to pass both = 25.

(b) Number who failed both = x – 25Max number who failed both = 30 – 25 = 5Min number who failed both = 25 – 25 = 0

(c) Number who passed Maths but failed Science = 30– x Max = 30 – 25 =5Min = 30 – 30 = 0

(d) x – 25 = 2, therefore x = 2727 students passed both tests.

7/27/2019 sets.pdf



Problem Solving -- Exercise 1-12

1. If n(A)=10, n (B)=6 and n(A∩B)=3, find n(A∪B).

2. In a group of 80 students, 40 play the piano and 30 play the guitar. If 20 students play neither instruments, how many play both piano and guitar?

3. In a group of students, 50% play the piano only and 10% play the guitar only. If 15% plays bothpiano and guitar, what is the percentage of students who play neither ?

4. In a class of 40 students, 30 passed Maths and 25 passed Science. What is the max andminimum no. of students who

(a) passed both Maths and Science(b) passed Maths but failed Science(c) failed both

More Examples 1-13

Example 1

Mr DoDo sells noodles. Given ε = {all noodles}, F = {noodles with fish}, E = {noodles with eggs}, M ={noodles with mushrooms}

(a) Express the following statements in set notations(i) All noodles with eggs have either fish or mushrooms.(ii) Some noodles with mushrooms have no eggs.(iii) Not all noodles with fish have eggs.

(b) Describe in words what noodles belong to the set:

(i) ( ) M E F ∩∪ '

(ii) ' E F ∩∩

Solution:

a) (i) M F E ∪⊆

(ii) ∅≠∩ ' E M

(iii) φ ≠∩ ' E F

b) (i) Noodles with mushrooms only.(ii) Noodles with fish and mushrooms only.

Example 2

Given that { } R x x xY ∈≤<= ,2011: and { },,135: R x x x Z ∈<≤= express each of thefollowing in similar set notation.

(i) Z Y ∪

(ii) Z Y ∩'

(iii) '' Z Y ∪

(iv) '' Z Y ∩

7/27/2019 sets.pdf



Solution:

(i) { } R x x x Z Y ∈≤≤=∪ ,205:

(ii) { } R x x x Z Y ∈≤≤=∩ ,115:'

(iii) { } R x xor x x Z Y ∈≥≤=∪ ,13,11:''

(iv) { } R x xor x x Z Y ∈><=∩ ,20,5:''

Exercise (Basic) 1-14

1. State whether each of the following statements is true (T) or false (F).

(i) If ∅= B AI , then ( ) ( ) ∅= B A B A IIU .

(ii) If ε = {rational numbers}, A = {x: x ≤ 7} and C = {4, 5, 6}, then ( ) B AC I⊆ (iii) If B B A =U , then A B ⊆ .

(iv) If A is a subset of B, then B B A =U .

(v) If A = {0, 1, 2, 0, 2, 2, 1, 1}, B = {2, 1, 0}, then A = B.(vi) If A = {rectangles with five vertices}, B = {odd numbers that are divisible by 12}, then A = B(vii) There are 32 subsets in the set {a, e, i, o, u}.

2. Of the 50 houses in the neighbourhood of Jalan Chantek, 32 have mango trees and 28 haverambutan trees. How many houses have both mango and rambutan trees if 4 houses haveneither?

3. (i) If n(A) = 80, n(B) = 25 and n( B AI )=10, find n( B AU ).

(ii) If n(P) =45, n(Q) =21 and ∅=Q P I , find n( Q P U ).

(iii) If n(R) =40, n(S) =72 and S R ⊆ , find n( S R I ).

4. Draw three intersecting sets as shown below for each of the following parts and shade the regionrepresenting each of the given set.

(i) ( ) Z Y X IU '

(ii) ( ) Y Z X UI '

(iii) ( ) Z Y X IU

(iv) (v) '' Z Y X II

(v) ( ) ( ) Z X Y X UIU

XY

Z

7/27/2019 sets.pdf



5. If ε = {integers from 1 to 13, both inclusive}, P = {prime numbers} and Q = {odd numbers}, list the

elements of the following.

(i) Q P I (ii) Q P U (iii) '' Q P U

(iv) '' Q P I (v) 'Q P I

6. Identify the set shaded in each of the following Venn diagrams.

(i)

(ii)

(iii)

(iv)

Exercise (Intermediate) 1-15

1. In a group of 30 children, the number of children who like apples is twice the number who likeoranges. If 3 like neither and 6 like both, how many like apples?

2. In a class of 40 children, 24 study Physics, 2 study Physics and Chemistry but not Biology, 8study Physics and Biology, 7 study Biology and Chemistry, 3 study Chemistry only and 2 study allthree subjects. If all the pupils study at least one of the three, find the number of children who

(i) study exactly one subject,(ii) study exactly two subjects,(iii) do not study Chemistry.

A

ε

P

P

ε

P Q

7/27/2019 sets.pdf



3. The Venn diagram represents subsets X, Y and Z of the universal set. On three separate copiesof the Venn diagram, shade the regions representing the following.

(i) Z Y X II

(ii) ( ) ( ) Z Y Y X IUI

(iii) ( ) X Z Y IU '

4. ε = {all triangles}, I = {isosceles triangles}, E = {equilateral triangles} and R = {right-angled

triangles}. Draw a Venn diagram to illustrate the relationship between , I, E and R.

5. Identify the set shaded in each of the following Venn diagrams.

(i) (ii)

________________________ ________________________

(iii) (iv)

________________________ ________________________

(v) (vi)

_________________________ ________________________

ε

X

Y

Z

ε

P Q

R

ε P

Q R

P Q

R

P Q

R

ε

Q

R

P P Q

R

7/27/2019 sets.pdf



Exercise (Advanced) 1-16

1. In the Venn diagram, ε is the set of all children in a certain chosen group,

ε

A = {children in Youth Club A} and B = {children in Youth Club B}.The letters p, q, x and y in the diagram represent the number of children in each subset.Given that n( ε ) = 220, n(A) = 85 and n(B) = 36,

(i) express p in terms of x ,

(ii) find the smallest possible value of y ,(iii) find the largest possible value of x ,(iv) find the value of q if p = 55.

2. The following table shows the test results of 40 students.

Subjects Pass Fail

Physics 34 6

Chemistry 32 8

Biology 35 5

It is given that x students passed all three subjects, 30 students passed both Biology and Physics,and 9 students passed exactly two subjects. It is also given that y students failed both Physics

and Chemistry, 2 students failed both Biology and Chemistry, 1 student failed both Biology andPhysics, and no student failed all three subjects.

(a) Taking P = {students who passed Physics},C = {students who passed Chemistry},B = {students who passed Biology},

illustrate the number of passes by means of a Venn diagram.

(b) Form two equations in x and y and find the values of x and y .

(c) If X = B’, Y = P’ and Z = C’,

(i) copy and complete the following Venn diagram to illustrate the number of failures.

(ii) explain how you would use this Venn diagram to find the value of x .

A B

p q

y

x

ε

X

YZ

x

1 20

6 – y

7/27/2019 sets.pdf



3. A survey of people shows that 35% watch SBC Channel 5, 42% watch Channel 8, 41% watchChannel 12, 20% watch none of the three channels, 16% watch Channels 5 and 8, 22% watchChannels 8 and 12 and 18% watch Channel 5 and 12. What percentage of the people watch(a) exactly two television channels,(b) Channel 12 only?

4. In this question,

ε = {pupils in a class},G = {girls},L = {pupils who pass Literature},H = {pupils who pass History},B = {pupils who pass Biology},M = {pupils who pass Mathematics}C = {pupils who pass Chemistry}

(a) Write sentences (not using technical words like ‘set’, ‘intersection’) to express the followingsymbolic statements.

(i) ∅=''' B H L II (ii) M C ⊂ (iii) L M L =I

(iv) C C B =U (v) ∅='G H I

(b) If all the statements in part (a) are true, name the subject in which no boy fails. Give reasons for

your answer.

Answers Key:

Practice 1

1.a) M = {April, June, September, November}b) G = { 2 , 3 , 5 , 7 , 11, 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 }c) H = {Africa, Antarctica, Asia, Australia, Europe, North America, South America}

2.

d) A = { 6 , 12 , 18 , 24 , 30 , 36 , 42 , 48 , 54 , 60 , 66 , 72 }e) B = { 2 , 4 }f) C = { -2 , -1 , 0 , 1 , 2 , 3 , 4 , 5 }

3.(a) ( T )

(b) ( T )

(c) ( F )

(d) ( T )

(e) ( F )

(f) ( T )

(g) ( F )

(h) ( F )

(i) ( T )

(j) ( T )

(k) ( T )

(l) ( T )

4.(a) ( T )

(b) ( F )

(c) ( F )

(d) ( F )

(e) ( F )

(f) ( T )

(g) ( T )

(h) ( T )

(i) ( T )

(j) ( F )

(k) ( T )

7/27/2019 sets.pdf



Practice 2

1.(a) x = {4}

(b) 7},6,5,4,3,2,1{= x

(c) { }30.....9,8,7,6= x

306 ≤≤ x

2.(a) A = { Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sept, Oct, Nov, Dec}

B = { Apr, Jun, Sept, Nov}C = { Feb}D = { Jan, Mar, May, Jul, Aug, Oct, Dec}E = { Feb, Apr, Jun, Sept, Nov}

(b) (i) ( T ) (ii) ( F )(iii) ( T ) (iv) ( F )(v) ( T ) (vi) ( F )(vii) ( F ) (viii) ( F )

3. (a) A ∩ B { }6= (b) A ∪ B { }8,6,5,4,3,2,1= (c) A ∩ C { }3,1= (d) B ∪ C { }8,6,4,3,2,1=

(e) A ∩ B ∩ C ∅= (f) A ∪ B ∪ C { }8,6,5,4,3,2,1=

4.(i)

(a) A ∩ B⇒⇒⇒⇒ Students who passed A Maths and Biology tests.

(b) C’⇒⇒⇒⇒ Students who failed Chemistry test.

(ii)(a) Nobody passed both A Maths and Biology tests.(b) All the Sec 3 students passed either their A Maths test or Chemistry test or both.(c) Students who failed Chemistry test, also passed A Maths test.

(d) Students who passed Biology test also passed Chemistry test.(e) All passed their A Maths test.

(iii)

(a) ( ) ( ) ε ε ==∅= C nC nC //'

(b) ε =∪∅=∩ C AC A /''

(c) AC ⊆

(d) ∅=∩∩ C B A

(e) ε =∪∪∅=∩∩ C B AC B A /'''

(f) ( ) 120=∩ B An

(g) ( ) 15'' =∩ BC n

(h)

( ) 32'=∩

BC n

7/27/2019 sets.pdf



Practice 3

1. A ∩ B'

(e) A ∪ B'

(f) A ∩ B ∩ C

(g) (A ∪ B) ∩ C

(h) A ∪ (B ∩ C)

2.

(a) ∅≠∩⊂ C B A B ,

(b) ∅≠∩⊂ C A A B , , ∅=∩C B

(c) ∅=∩ B A , ∅≠∩C A , ∅=∩C B

(d) ∅=∩∅≠∩∅≠∩ C AC B B A ,,

3. (a) ( )'C B A ∪∩

(b) ( ) ( ) ( )C BC A B A ∩∪∩∪∩

(c) ( )'C B∪

(d) C A∪

BA

A BBA

A B

C

BA

C

BA

C

A B

C

BA

C

A B

C

7/27/2019 sets.pdf



4.

5.

Problem Solving Exericse

1. 13 2. 10 3. 25%4(a) 25, 15 4(b) 15, 5 4(c) 10,0

Exercise (Basic):

1. (i) T 1. (ii) T 1. (iii) F1. (iv) T 1. (v) T 1. (vi) T1. (vii) T 1. (viii) T 1. (ix) F2. 29 3. 95, 66, 40

4.

(i) d ( ) Z Y X IU ' (ii) ( ) Y Z X UI '

(iii) ( ) Z Y X IU (iv) '' Z Y X II

I

E

R

ε

A

ε

A

B 4

E

C 1

D 7

ε

X Y

Z

ε

X Y

Z

ε

X Y

Z

7/27/2019 sets.pdf



(v) ( ) ( ) Z X Y X IU ∩

5. (i) { }13,11,7,5,3=Q P I 5. (ii) { }13,11,9,7,5,3,2,1=Q P U

5. (iii) { }12,10,9,8,6,4,2,1'' =Q P U 5. (iv) { }12,10,8,6,4'' =Q P I

5. (v) { }2'=Q P I 6. (i) 'Q P ∩

6. (ii) ( ) ''/' Q P Q P IU 6. (iv) )'( Q P ∩

Exercise (Intermediate):

1. 22 2. 25, 13, 28

3.(ii) (iii)

4.

5. (i) Q P U 5. (ii) R P I 5. (iii) ( ) Q R P IU

5. (iv) 'Q R P II 5. (v) ( )' R P Q UI 5. (vi) ( ) ( ) ( ) R P RQQ P IUIUI

Exercise (Advanced):

1. (i) x p −= 85 1. (ii) 99

1. (iii) 36= x 1. (iv) 6,30 == q x

ε

X Y

Z

ε

X

Y

Z X

Y

Z

E

I

R

7/27/2019 sets.pdf



2. (a)

2. (b) 26= x , 2= y

(c) (i)

3. (a) 2% 3. (b) 19%4. (a) (i) No one failed all the 3 subjects, Literature, History and Biology.4. (a) (ii) All who pass Chemistry also pass Mathematics.4. (a) (iii) All who pass Literature also pass Mathematics.4. (a) (iv) All who pass Biology also pass Chemistry.4. (a) (v) No boys pass History.4. (b) No boys pass History.⇒They pass Literature / Biology.⇒ Those who pass Biology, pass

Chemistry and also pass Mathematics. Those who pass Literature also pass

Mathematics.⇒Hence, all boys passed Mathematics.

ε B P

C

30 – x

2 y

2

x

5 –

1

ε

X

YZ

x

1 20

6 – y5 – y

(40)(5)

(6)

2

(8)

Students who pass all 3

Students who failed all 3

sets.pdf

Documents