21 - edupub.gov.lk 21.pdf · 21 21.1 Finite and infinite sets. If the number of elements in a set is finite that set is said to be a finite set. ... 21.2 Subset and the complement

Sets21

21.1 Finite and infinite sets.

If the number of elements in a set is finite that set is said to be a finite set. If the number of elements in a set is infinite that set is said to be an infinite set.

For Free Distribution53

1. Of the sets given below, indicate the finite sets and infinite sets separately.P = {Vowels of the English alphabet} T = {The subjects you study in school}Q = {Prime numbers} U = {Positive numbers less than 100}R = {Polygons} V = {Triangular numbers}S = {Colours of the rainbow}

2. (i) Write 5 examples for finite sets(ii) Write the number of elements in each set written above.

3. Write 5 examples for infinite sets.

Teacher - There are 32 children in this class. How many of them are doing music?

Sumudu - Teacher, there are thirteen students.Teacher - Ah! How many of them are playing musical instruments ?Sumudu - Only three teacher. The other ten are singing.Teacher - Good, then all of you come prepared to stay after school for practices.Nadeesha - As I belong to the "complement", I do not have to stay after school.Sumudu - What do you mean by the "complement"Nadeesha - When our whole class is considered as the universal set, the students

who are doing music is a set consisting of some elements of the universal set. Then those who are not doing music will belong to the complement of that set.Now I enclose all the students in our class inside this square and draw a circle enclosing those who are doing music. Those who are playing the instruments are also in this circle. So I enclose them in this small circle. You are also in that small circle. I will be outside the big circle, but inside the square.

Sumudu - A fine diagram.Nadeesha - Not only that, the sets inside another set are

called subsets.{Children who are doing music}is a subset of the set {Children in our class}.The set of those who are playing instruments is a subset of the set of children doing music.

Sumudu - Ah! What a lot you know Nadeesha!

21.2 Subset and the complement of a set.

A set which consists of elements which belong to the universal set but do not belong to a given set is called the complement of this given set

Those who aredoing music

Those who areplaying musicalinstruments

S

N

Students in the class

Exercise 21.1


S - SumuduN - Nadeesha

The complement of A is denoted by . It can be expressed in a Venn diagram by shading as follows.

e = {1, 2, 3, 4, 5, 6}A = {3, 6}

/The complement of A , A = {1, 2, 4, 5}In a Venn diagram it can be shown as follows.

If B is a set formed by several elements of A then B is a subset of AIt is denoted by B Ì A

It can be shown in a Venn diagram as follows.

P = {1, 2, 3, 4, 5, 6, 7}Q = {2, 4, 6}R = {3, 6, 9}

All the elements of Q belong to P Q is a subset of P. i.e., Q Ì PBut 9 does not belong to P (9 Ï P)\ R is not a subset of Pi.e., R Ë P

P and Q are illustrated in the Venn diagram.

e = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}X = {Prime numbers between 1 and 10}Y = {Multiples of 3 less than 10}

/(i) Write the set X /

(ii) Write the set Y (iii)Write 5 subsets of X (iv)Write all the subsets of Y

/A

Example -

Example -

A set which has no elements, several elements, or all the elements of a certain set is called a subset of the original set.

Example 1

e

A/

A

eA

/

A

5

21

4

AB

5 42 6

7

3

QP

1

36


a c

d

f

be

BA

e

(i) X = {2, 3, 5, 7} \ x = {1, 4, 6, 8, 9, 10}(ii)Y = {3, 6, 9}

/\ Y = {1, 2, 4, 5, 7, 8, 10}

(iii){2}, {3}, {5}, {7}, {2, 3}, {2, 5}, {2, 7}, {3,5}, {3,7}, {5, 7}, {2,3,5}, {2,3,7}, {2,5,7}, {3,5,7}, {2,3,5,7}, {} are the subsets of X.Any five sets out of these sets can be taken as the answer

(iv){3}, {6}, {9}, {3, 6}, {3, 9 },{6, 9 },{3, 6, 9 },{ }

e = {a, b, c, d, e, f}A = {a, c, e}B = {a, c}

(i) Illustrate the above sets in aVenn diagram/

(ii) Write the set A /(iii) Write the set B

(iv) Write the relationship between the sets A and B using set notation.

(i) As all the elements of B belong to A, the set B is inside the set A

/ (ii) A = {b, d, f}/

(iii) B = {b, d, e, f}(iv) B Ì A

/ 1. Write the complement (A ) of the set A in each of the following.

(i) e = {Children in our class}A = {Girls in our class}

(ii) e = {Counting numbers between 1 and 10}A = {Even numbers between 1 and 10}

(iii)e = {The farmers in our village}A = {The farmers who grow paddy}

/

The null set is a subset of any set { }Ì A

Every set is a subset of itself A Ì A

«

«

Example 2

Exercise 21.2


(iv)e = {Passengers in a bus}A = {Passengers in the bus who are carrying umbrellas}

(v) e = {People attending the lecture}A = {People attending the lecture who are more than 50 years of age}

(vi)e = {Children who participated in the tour}A ={The boys who participated in the tour}

2. e = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}P = {5, 10}Q = {2, 4, 6, 8}

/(i) Write P and Q (ii) Write all the subsets of P (iii) How many subsets does Q have?

3. Copy the following expressions. If they are correct mark them with ( ü ) and if they are wrong mark them with (´)

(i) {5}Ì {Prime numbers}

(ii) {3, 5}Ì {Counting numbers}(iii){0, 2, 3}Ì {Digits in the number 20 421}

(iv){R}Ë {The letters in the word " RATHNAPURA"}

(v) {1}Ì {Triangular numbers}(vi){January}Ì{The months with only 30 days}(vii){0, 1, 4, 5}Ì {The digits in the units place of the perfect square numbers}

(viii) 4 Ì {Even numbers}/4. (a) n(A) + n(A) = n(e). Verify this statement using Venn diagrams

/(b) (i) If n(e) = 12 and n(A) = 7 find n(A)

/(ii) If n(X) = 20 and n(X ) = 13 find n(e)

/(iii) If n(e) = 35 and n(P ) = 18 find n(P)

5. A = {2}, B = {2, 3}, C = {2, 3, 5}, D = {2, 3, 5, 7}

(i) Copy and complete the table given below.

(ii) If the number of elements in a set is n, write the number of subsets of that set in terms of n based on the above table.

/

Set Number of elements

Subsets Number of subsets

Number of subsets as a power of 2

A --------- ------------------------ -------- ----------2B 2 {2}, {3}, {2, 3}, {} 4 2

C -------- ------------------------ --------- ----------D -------- ------------------------ -------- ----------


S = {Pupils from grade 9 who have membership in the school sports club}R = {Pupils from grade 9 who are members of the scouts association }

By listing the elements of the above sets the following results are obtainedS = {Thilini, Manosha, Udayangani, Chatura, Dinidu, Yohan}R = {Udayangani, Nalaka, Chamara, Isuru, Dinidu, Nelum, Nethmini}

According to the above information, Udayangani and Dinidu belong to both S and R. Hence the set {Dinidu, Udayangani}is equal to the set {Students from grade 9 who are members of the school sports club and the scouts association}

The intersection of S and R is,S Ç R = {Udayangani, Dinidu}(This is read as S intersection R)

When all the students of S and R are considered, it is the set {students from grade 9 who are members of the school sports club or the scouts association}

The union of S and R is,

S È R = {Thilini, Manosha, Udayangani, Chatura, Dinidu, Yohan, Nalaka, Chamara, Isuru, Nelum, Nethmini}

(This is read as S union R)

The two sets can be represented in a Venn diagram as follows.

Elements in the intersection show the properties of both sets.

P = {Quadrilaterals}Q = {Regular polygons} P Ç Q = {Regular quadrilaterals}

Example -

21.3 Intersection and union of sets

The set of elements common to two sets is called the intersection of the two sets and is denoted by the symbol Ç

The set containing all the elements of two sets is called the union of the two sets and is denoted by the symbol È

ThiliniManoshaChaturaYohan

UdayanganiDinidu

NalakaChamaraIsuruNelumNethmini

S R


P = {Digits in the number 534 063}Q = {Digits in the number 120 347}R = {Digits in the number 217 891}

(i) List the elements of each set given above.(ii) List the elements of P Ç Q (iii) List the elements of P Ç R(iv) List the elements of P È Q(v) List the elements of Q È R

(i) P = {5, 3, 4, 0, 6} Q = {1, 2, 0, 3, 4,7} R = {2, 1, 7, 8, 9}

If e = {Counting numbers between 0 and 16}X = {Multiples of 4 between 0 and 16}Y = {Multiples of 3 between 0 and 16}

(a) (i) List the elements of the above sets.(ii) Show these sets in a Venn diagram.

(b) List the elements of the following sets./ / /(i) X Ç Y (ii) X È Y (iii) X (iv) (X È Y) (v) X Ç Y

(a) (i) e = {1,2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}X = {4, 8, 12}Y = {3, 6, 9, 12, 15}

(ii)

(b) (i) X Ç Y = {12}(ii) X È Y = {3, 4, 6, 8, 9, 12, 15}

/(iii) X = {1,2, 3, 5, 6, 7, 9, 10, 11,13, 14, 15}

/(iv) (XÈY) = {1,2, 5, 7, 10, 11,13, 14}

/(v) Here you have to take the elements common to X and Y

/\ X Ç Y = {3, 6, 9, 15}

Example 3

Example 3

10 11 1314

12

15

34

8

3

9

7521

YX

e

6

(ii) P Ç Q = {3, 4, 0}(iii)P Ç R = { } (iv)P È Q = {5, 3, 4, 0, 6, 1, 2, 7}(v) Q È R = {1, 2, 0, 3, 4, 7, 8, 9}


Exercise 21.3

If A = {1, 3, 6, 10, 15, 21, 28, 36, 45}B = {6, 12, 18, 36, 42, 48}C = {3, 6, 9, 12, 15, 18, 21}, write the following sets.(i) A Ç B (ii) A Ç C (iii) B Ç C (iv) A È B (v) B È C

2. If e = {Counting numbers from 1 to 10} P = {digits of the number 435 308 105} Q = {prime factors of 180}

R = {prime factors of 196}

(a) List the elements of the above sets(b) List the elements of the following

(i) P Ç Q (ii) Q Ç R (iii) P Ç R (iv) P È Q (v) Q È R/(vi) P Ç Q (vii) P Ç (Q È R) (viii) (P È Q) Ç (P È R)

3. Write the following sets using the Venn diagram(i) A (ii) B(iii) A Ç B (iv) A È B

/ /(v) (A È B) (vi) (A Ç B)

/ /(vii) A (viii) B / /(ix) A Ç B (x) B Ç A

4. For the pairs of sets given below write the following sets, in words.

(i) A Ç B (ii) A È B

(a) A = {Grade 9 students in Tissa Vidyalaya}B = {Students in the prefects board of Tissa Vidyalaya}

(b) A = {Students of Gamunu Vidyalaya who play net ball}B = {Students of Gamunu Vidyalaya who play volley ball}

(c) A = {Students in our school who have visited Dambulla}B = {Students in our school who have visited Sigiriya}

(d) A = {The members of the Aruna Farmers association who grow vegetables}B = {The members of the Aruna Farmers association who grow paddy }

(e) A = {Students in our school who could speak in English}B = {Students in our school who could speak in Tamil}

10 35

15

30

40

504535

BA

e

25

20


5. The marks obtained by a group of students at written tests in Mathematics and Science are given below in a table.

If S = {Students who obtained more than 90 for Science}M = {Students who obtained more than 90 for Mathematics}

(i) Write the sets S and M (ii) Write the set S Ç M (iii)Write the set S È M (iv) An institution named 'A' offers

scholarships to the students who have obtained more than 90 marks for Science and Mathematics. Who are the students who have qualified for this scholarship?

(v) Another institution named 'B' offers scholarships to the students who have obtained more than 90 for Science or Mathematics. Who are the students who have qualified for this scholarship?

6. Copy the Venn diagrams and show the set given below in each Venn diagram by shading the relevant region.

Nishani 68 85Sadaruwan 87 96Buddhi 94 95Nayomi 89 92Nadeeshani 95 97Deleepa 82 74Aruni 93 79Malan 83 71

Marks

Name Science Maths

BA

e

Y

B

X

A

e

e

P

e

Q

Q

P

P

e

e

P / A Ç B

X Ç Y/

A È B P Ç Q

P È Q

(i) (ii)

(iii) (iv)

(v) (vi)


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21 - edupub.gov.lk 21.pdf · 21 21.1 Finite and infinite sets. If the number of elements in a set is finite that set is said to be a finite set. ... 21.2 Subset and the complement