Cardinal Numbers: Overview, Questions, Preparation

Sets are nothing but a collection of objects. For example, the collection of the first five natural numbers represents a set {1,2,3,4,5}.
One-to-one correspondence and equivalence of sets: 
The condition in which the elements of any two sets are paired such that each element is paired with exactly one element from the other set is known as one-to-one correspondence.

Cardinal number of a set:
Cardinal number is nothing but counting numbers. 
Definition: The number of distinct elements in a set is known as its cardinality. It is represented as n(X) where X is a set and is read as "the number of elements in set X."
Example:
Set X= {11, 10, 14, 3}
There are 4 elements in the set, and they all are distinct. Hence, n(X) = 4.

Image Source: NCERT

The cardinality of a finite set is always a natural number. It can never be a fraction or anything else.
Two sets are said to have the same cardinality if they show a one-to-one correspondence between them. 

Example:

Set X= {P, Q, R}

Set Y= {S, T, U}

we know that these two sets are one-to-one correspondence and equivalent. Now, set X has three elements (all distinct) and set Y also has three elements (again all distinct).
n(X) = 3 and n(Y) = 3. Set X and set Y have the same cardinality. 

Importance and Weightage

This chapter is taught in Class 11th.  In this chapter, you will study concepts and definitions regarding sets. Moreover, you will get to know how to represent a set, types of sets, subsets, Venn diagrams, laws of the algebra of sets. Sets are of 8 marks.

1. Write the following sets in roster form:
(i) A = {x : x is an integer and –3 (ii) B = {x : x is a natural number less than 6}

Solution. (i)Elements are: -2, -1, 0, 1, 2, 3, 4, 5, 6
roster form is:
A = {-2, -1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {x : x is a natural number less than 6}
Elements are:  1, 2, 3, 4, 5
roster form is:
B = {1, 2, 3, 4, 5}

2. Let V = { a, e, i, o, u } and B = { a, i, k, u}. Find V – B and B – V.

Solution.

V-B= {e, o} since a, i and u belong to both the sets and the elements e and o belong to V but not to B.

B-V= {k} since k belongs only to set B and all the other elements present in B are also present in V.

3. If S and T are two sets such that S has 21 elements, T has 32 elements, and S ∩ T has 11 elements, how many elements does S ∪ T have?

Solution. Given, 

n(S) = 21

n(T) = 32

n(S∩T) = 11 

n( S ∪ T) = n(S) + n(T) – n(S∩T)

n( S ∪ T) = 21+ 32+ 11

n( S ∪ T) = 42

Therefore, ( S ∪ T) has 42 elements.

Q: Are all equivalent sets equal?

Q: What is the difference between an equal set and an equivalent set?

Q: What is the power of a set, say X?

Q: If X is a subset of a universal set U, then what is one possible relation between the complement of A and U?

Q: What is the cardinal number of an empty set?