ICSE Class 6 Maths Chapter 29 Cardinal Property of a Set

Chapter 29: Cardinal Property Of A Set

Cardinal Number Of A Set

The number of elements in a set is called its cardinal number.

For example:

(i) Set P = { 2, 9, 11, 14 } has 4 elements. Therefore, Cardinal number of set P = 4.

(ii) Set M = { x, y, z } has 3 elements. Therefore, Cardinal number of set M = 3.

(iii) Set E = { } has no element. Therefore, Cardinal number of set E = 0 and so on.

The symbol used for showing the cardinal number is the small letter 'n' attached before the name of the set that is written inside brackets.

Thus, the cardinal number of set A is represented by n(A).

Conversely, n(B) represents the cardinal number of set B.

Thus, for the sets P, M and E given above:

(i) n(P) = 4, as set P has only four elements; 2, 9, 11 and 14.

(ii) n(M) = 3, as set M has only three elements; x, y and z.

(iii) n(E) = 0, as set E is the empty set and so on.

Note: Cardinal number of the empty set 0. Cardinal number of infinite set is not defined.

Example 1

Write the cardinal number of each of the following sets:

(i) A = { 2, 3, 5, 5, 3, 3 }

(ii) B = { letters in the word "NOORJAHAN" }

(iii) P = { counting numbers between 10 and 30; that are divisible by 5 }

Solution

(i) Since A = { 2, 3, 5, 5, 3, 3 } = { 2, 3, 5 }, Cardinal number of set A = 3, i.e. n(A) = 3

(ii) Since B = { n, o, r, j, a, h }, Cardinal number of set B = 6, i.e. n(B) = 6

(iii) Since P = { 15, 20, 25 }, n(P) = 3

Example 2

Let A = { 5, 6, 8, 9 }, B = { 3, 6, 9, 12, 15 } and C = { 2, 4, 6, 8, 10, 12 }. Find:

(i) n(A)

(ii) n(A ∪ B)

(iii) n(B ∩ C)

(iv) n(B) + n(A ∪ C)

Solution

(i) Set A has 4 elements; therefore n(A) = 4

(ii) Since A ∪ B = {3, 5, 6, 8, 9, 12, 15}, therefore n(A ∪ B) = 7

(iii) Since B ∩ C = {6, 12}, therefore n(B ∩ C) = 2

(iv) Set B has five elements, therefore n(B) = 5. Since A ∪ C = {2, 4, 5, 6, 8, 9, 10, 12}, therefore n(A ∪ C) = 8 and n(B) + n(A ∪ C) = 5 + 8 = 13

Teacher's Note

Understanding cardinal numbers helps in real-world situations like counting inventory in a store or determining how many people are in a group.

Exercise 29

1. Write the cardinal number of each of the following sets:

(i) A = { 0, 1, 2, 4 }

(ii) B = { - 3, - 1, 1, 3, 5, 7 }

(iii) C = { }

(iv) D = { 3, 2, 2, 1, 3, 1, 2 }

(v) E = { Natural numbers between 15 and 20 }

(vi) F = { Whole numbers from 8 to 14 }.

2. Given: A = { Natural numbers less than 10 }, B = { Letters of the word 'PUPPET' }, C = { Squares of the first four whole numbers }, D = { Odd numbers divisible by 2 }

Find:

(i) n(A)

(ii) n(B)

(iii) n(C)

(iv) n(D)

(v) A ∪ B and n(A ∪ B)

(vi) A ∩ C and n(A ∩ C)

(vii) n(B ∪ D)

(viii) n(C ∩ D)

(ix) n(B ∪ C)

(x) n(A ∪ D).

3. If A = { 5, 7, 8, 9 }, B = { 3, 4, 5, 6 } and C = { 2, 4, 6, 8, 10 }; find:

(i) n(A) + n(B)

(ii) n(A ∪ B)

(iii) n(A ∩ B)

(iv) n(A ∪ B) + n(A ∩ B)

(v) n(B ∪ C)

Is n(A) + n(B) = n(A ∪ B) + n(A ∩ B)?

4. Given: P = Set of letters in the word 'ALLAHABAD' and Q = Empty set,

find:

(i) n(P)

(ii) n(Q)

(iii) n(P ∪ Q)

(iv) n(P ∩ Q)

Is n(P) = n(P ∩ Q)?

Is n(Q) = n(P ∪ Q)?

Is n(P) = n(P ∪ Q)?

5. State true or false for each of the following. Correct the wrong statement.

(i) If A = { 0 }, then n(A) = 0.

(ii) n(∅) = 1.

(iii) If T = { a, l, a, h, b, d, h }; then n(T) = 5

(iv) If B = { 1, 5, 51, 15, 5, 1 }, then n(B) = 6.

Teacher's Note

These exercises reinforce the concept that duplicate elements in a set are counted only once, similar to how we don't count the same book twice when organizing a library shelf.

Revision Exercise (Chapter 29)

1. For each of the following statements, write true or false.

(i) If n(A) = n(B), sets A and B are equal.

(ii) If sets A and B are equal, n(A) = n(B).

(iii) If set A = { 3, 4, 4, 4, 3, 2, 7 }, n(A) = 4.

(iv) If set B = { 5, 5, 5, 4 }, n(B) = 4.

(v) If P = set of natural numbers less then 6, n(P) = 6.

(vi) If M = set of whole numbers between 10 and 15, n(M) = 4.

2. If A = { 5, 6, 7, 8, 9 } and B = { 7, 8, 9, 10 }

Find: (i) A ∪ B and n(A ∪ B)

(ii) A ∩ B and n(A ∩ B)

3. If n(D) = 0, set D is the empty set. Is this statement true?

4. If M = { 2, 4, 6, 8, 10, 12 } and N = { 4, 8, 12, 16, 18 }

Find: (i) M ∪ N

(ii) M ∩ N

(iii) n(M) and n(N)

(iv) n(M ∪ N) and n(M ∩ N)

(v) n(M) + n(N)

(vi) n(M ∪ N) + n(M ∩ N)

5. If A = { 10, 20, 30, 40 } and B = The empty set.

Find: (i) A ∪ B

(ii) A ∩ B

(iii) n(A ∪ B)

(iv) n(A ∩ B)

6. If set P = {x : x is a natural number between 15 and 23}, write:

(i) set P in roster form

(ii) cardinal number of set P.

7. If set M = {x : x is a whole number less than 8}, write:

(i) set M in tabular form

(ii) cardinal number of set M.

8. If set S contains all values of x where x is a whole number and 5 ≤ x < 15, write:

(i) set S in roster form

(ii) cardinal number of set S.

Teacher's Note

These revision exercises help reinforce understanding of cardinal numbers through various set operations, much like how taking inventory teaches us to count and organize items systematically.

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