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Cardinality

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Question

Write the cardinal number of each of the following sets:

$G = {x : x x is a perfect square number, x \in N a n d x \leq 30}$

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Solution

Step : Finding the cardinal number of the given set

Given:

$G = {x : x x is a perfect square number, x \in N a n d x \leq 30}$
Perfect square natural numbers less than 30 are 1, 4, 9, 16, 25 i.e., 5 elements.

We know that the cardinal number of a set is the number of elements in it.
Hence, the cardinal number is 5.

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Q. Taking the set of natural numbers as the universal set, write down the complements of the following sets: (i) { x : x is an even natural number} (ii) { x : x is an odd natural number} (iii) { x : x is a positive multiple of 3} (iv) { x : x is a prime number} (v) { x : x is a natural number divisible by 3 and 5} (vi) { x : x is a perfect square} (vii) { x : x is perfect cube} (viii) { x : x + 5 = 8} (ix) { x : 2 x + 5 = 9} (x) { x : x ≥ 7} (xi) { x : x ∈ N and 2 x + 1 > 10}

Find the cardinal number of the following sets :

(i) $A_{1} = {- 2, - 1, 1, 3, 5}$

(ii) $A_{2} = {x : x \in N a n d 3 \leq x < 7}$

(iii) $A_{3} = {p : p \in W a n d 2 p - 3 < 8}$

(iv) $A_{4} = {b : b \in Z a n d - 7 < 3 b - 1 \leq 2}$ $Cardinal~Number~of~set~:The number of elements in a set is called its Cardinal Number.$

Q. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) {

x : x

is an even natural number}

x : x

is an odd natural number}

x : x

is a positive multiple of

3

x : x

is a prime number }

x : x

is a natural number divisible by

3

5

x : x

is a perfect square }

x : x

is a perfect cube}

{x : x + 5 = 8}

{x : 2 x + 5 = 9}

{x : x \geq 7}

{x : x ϵ N a n d 2 x + 1 > 10}

Q. Find the cardinal number of the following sets
(i)

A =

{

x : x = 5^{n}, n \in N

n < 5

}

B =

{

x : x

is a consonant in English Alphabet

}

X =

{

x : x

is an prime number

}

P = {x : x < 0, x \in W}

Q = {x : - 3 \leq x \leq 5, x \in Z}

Write the cardinal number of each of the following sets:

$C = {x : x \in N a n d x \leq 7} .$

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