How To Find Cardinal Number Of A Set

In this affiliate we will learn the definition of cardinal number of prepare with method of calculation and solved example.

What is Cardinal number of set?

The number of distinct element present in a gear up is called cardinal number of set up.

For Example;
Consider the fix A with post-obit elements.
A = { 2, five, vii, 9 }

Annotation that there are iv distinct element in the set A.
Hence the cardinal number of set A is 4.

Case 02
Consider the ready B with following elements.
B = { four, 7, x, 10 }

Here the number of elements nowadays in set up B is 4.
But annotation that the element 10 is repeated twice in the set up.

Really there are 3 distinct element in the above ready B.
Hence, the central number of set B is three.

Representing cardinal number in Set Theory

The key number is represented by following expressions;

For Case, Consider the beneath set A;
A = { 3, 5, 7, nine, 11 }

The cardinal number of set A is expressed as;

Special Instance of Primal Number

There are ii special cases of cardinal numbers which you need to understand.

(i) Cardinal Number of Empty set up
(2) Cardinal Number of Infinite set

Primal Number of Empty Fix

Given beneath is the empty set A.

A = { 𝜙 }

Note that in empty set there is no elements nowadays. Hence the cardinal number of empty gear up is 0.

Conclusion
The key number of empty set is 0

Cardinal Number of Infinite set

Space set is the set up which contains unending number of elements.

Given below is the fix A with infinite elements;
A = { ten : x is set of positive even numbers }

In Roster Form, the fix A tin can be written as;
A = { 2, 4, six, viii, ten, 12, . . . . . }

You lot tin encounter that the set is never ending, so the cardinal number cannot be defined for this fix.

Determination
The cardinal number of infinite set is not divers.

Solved Bug on Cardinal Number of Fix

(01) Given beneath are two gear up A & B.

Find the cardinal number of both set A & B

Solution
A = { ii, 4, vi, 6, 12, iii, three, 3 }

Notation that in that location are 8 elements in the above set.
But chemical element 6 is repeated twice and number 3 is repeated thrice.

Rewriting the set A with unique elements we get;
A = { 2, 4, six, 12, 3 }

There are five distinct elements in the set A.
Hence, the fundamental number of ready A is 5.
n (A) = 5

B = { half dozen, ane, 9, 3, half dozen }

There are v elements in gear up B.
But annotation that element 6 is repeated twice.

Rewriting the gear up B with distinct elements.
B = { 6, 1, 9, 3 }

So there are four distinct elements in set B.
Hence, the cardinal number of set B is iv.
due north ( B ) = 4

(02) Given below are ii sets Ten & Y.

Detect the fundamental number of both the sets.

Solution
10 = { M, A, T, H, E, Chiliad, A, T, I, C, S }

There are 11 elements in the set 10.
But there are elements which are repeated more than one.

Letter " M" is repeated twice.
Latter " A " is also repeated twice.
Similarly other messages also repeated in the ready X.

Rewriting the set X with distinct elements.

X = { M, A, T, H, Eastward, I, C, S }

Note that there are viii distinct elements in the to a higher place set.
Hence, the cardinal number of set 10 is 8.
n (X) = 8

Y = { x : 10 contain letters in word "Apple tree" }

Writing the prepare Y in Roster class
Y = { A, P, P, L, E }

Notation that in that location are 5 elements in the higher up set, but some of the letters are repeated.

Rewriting the gear up Y with singled-out elements, we get;
Y = { A, P, L, E }

Hence at that place are 5 distinct elements in set Y.
The primal number of set Y is 5.
n ( Y ) = 5

(03) Find the central number of below set up A.
A = { ten : natural number between 4 & 46 & divisible by five }

Solution
Let us kickoff write the prepare A in Roster form.
A = { five, 10, xv, twenty, 25, xxx, 35, 40, 45 }

Notation that there are ix distinct elements in set A.
Hence central number of gear up A is 9.
n (A) = 9

(04) Find the cardinal number of set X
X = { Africa, Due north America, Europe, Asia, Antarctica, South America }

Solution
There are 6 distinct element in set Ten.
Hence the cardinal number of set X is 6.
n (X) = 6

(05) Find the primal number of set A & B

Solution
(i) A = { x : 10 is prime number number between 4 & 15 }

Writing the prepare A in Roster form
A = { 5, 7, eleven, 13}

There are 4 distinct element in set up A.
Hence key number of gear up A is four.
n (A) = 4

(ii) B = { x : x is factor of number two}

Writing the prepare B in Roster form.
B = { 1, two }

There are two distinct element in set B.
Key number is set B is ii.
north (B) = two

Source: https://wtskills.com/cardinal-number-of-set/

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