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 “****a****∈****A****”; if ****b ****is not an element of ****A**** then we say that “****b ****does not belong to ****A****” and represent it as“****b****∉****A****”.**

- Description method
- Roster method or listing method or tabular form
- 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:**

- The order of listing the elements in a set can be changed.
- If one or more elements in a set are repeated, then the set remains the same.
- 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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