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Sets

Sets



In our daily lives, we talk about different collections such as collection of maths books in a cupboard, collection of toys in a shop, collection of shirts in a shop, students in a school, collection of all natural numbers, etc.

These collections are said to be sets.

A set is a well-defined collection of objects.

Sets are usually represented by capital letters A, B, C, D, X, Y, Z, etc. The objects inside a set are called elements or members of a set. They are denoted by small letters a, b, c, d, x, y, z, etc.

Now, let us consider the set of natural numbers. We know that 4 is a natural number. However, −1 is not a natural number. We denote it as 4 ∈ N and −1 ∉ N.

If a is an element of a set A then we say that “a belongs to A” and mathematically we write it as “aA”; if b is not an element of A then we say that “b does not belong to A” and represent it as“bA”.

There are three different ways of representing a set:
  1. Description method
  2. Roster method or listing method or tabular form
  3. Set-builder form or rule method

Let us study about them one by one.

Description method: In this method, a description about the set is made and it is enclosed in curly brackets { }.

For example: The set of composite numbers less than 30 is written as

{Composite numbers less than 30}

Roster method or listing method or tabular form : In the roster form, all the elements of a set are listed in such a manner that different elements are separated by commas and enclosed within the curly brackets { }. The roster form enables us to see all the members of a set at a glance.

For example: A set of all integers greater than 5 and less than 9 will be represented in roster form as {6, 7, 8}. However, it must be noted that in roster form, the order in which the elements are listed is immaterial. Hence, the set {6, 7, 8} can also be written as {7, 6, 8}.

Set-builder form or rule method: In set-builder representation of a set,all the elements of the set have a single common property that is exclusive to the elements of the set i.e., no other element outside the set has that property.

We have learnt how to write a set of all integers greater than 5 and less than 9 in roster form. Now, let us understand how we write the same set in set-builder form. Let us denote this set by L.

L = {x : x is an integer greater than 5 and less than 9}

Hence, in set-builder form, we describe an element of a set by a symbol x (though we may use any other small letter), which is followed by a colon (:). After the colon, we describe the characteristic property possessed by all the elements of that set.

Note:

  1. The order of listing the elements in a set can be changed.
  2. If one or more elements in a set are repeated, then the set remains the same.
  3. Each element of the set is listed once and only once.

Now, consider the following three sets.

A = {x: x Z, −18 < x ≤ 5}

B = {x: x W}

C = {x: x N, −7 < x < −1}

Did you observe anything about the number of elements of these sets?

Observe that if we count the elements of set A, then we find that the number of elements is limited in this set. However, the number of elements in set B is not limited and we cannot count the number of elements of this set. Also, observe that set C does not contain any element as there does not exist any natural number lying between −7 and −1.

Therefore, on this basis i.e., on the basis of number of elements, the sets are classified into following categories:

(a) Finite set

(b) Infinite set

(c) Empty set

(d) Singleton set

Let us now study about them one by one.

(a) Finite set − A set that contains limited (countable) number of different elements is called a finite set.

(b) Infinite set − A set that contains unlimited (uncountable) number of different elements is called an infinite set.

(c) Empty set − A set that contains no element is called an empty set. It is also called null (or void) set. An empty set is denoted by Φ or {}. Also, since an empty set has no element, it is regarded as a finite set.

(d) Singleton set − A set having exactly one element is known as singleton set.

Therefore, we can now classify the above discussed sets as follows:

A = {−17, −16, −15, …, 0, 1, 2, 3, 4, 5} → Finite set

B = {0, 1, 2, 3, 4, 5 …} → Infinite set

C = Φ or {} → Empty set

D = {4} → Singleton set

Now, again consider set A. We have seen that it is a finite set.

Can you find the number of elements in this set?

We have A = {−17, −16, −15, −14, −13, −12, −11, −10, −9, −8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5}

We see that the number of elements in set A is 23. This number 23 is known as the cardinal number of set A.

Cardinal number of a set is defined as:

The number of distinct elements in a finite set A is called its cardinal number. It is denoted by n (A).

Note: The cardinal number o...

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