Sets: Definition, Types, Representation & Class 11 Maths Notes

A set is a collection of objects, elements, or members. It can only be a set when the objects are well-defined. This is how mathematicians define sets. 

Usually, we denote sets in capital letter form. The elements are in small letters such as $a,b,c$ etc.

The most basic way of writing a set is below.

If an element a belongs to a set $A$ , we write it as $a\in A$ . If it does not belong, is a $\notin \mathrm{A}$ .

Importance of Sets (for JEE Mains)

• Sets are integrated into relations and functions - an important topic cluster for JEE Mains. Even after the National Testing Agency has reduced subjects, sets is still there. 
• With sets, you can understand advanced maths concepts, and this knowledge can help with advanced problem-solving in the JEE exams, and other engineering exams too. 

In chapter 1 of your NCERT Maths textbook for Class 11, the definition of set is

"A set is a well-defined collection of objects."

So, why does it have to be 'well-defined'? 

Your NCERT book quotes various examples to justify this.

One is where it talks about the rivers of India. The Ganges, the Yamuna, etc., are all part of it. So we can say that these rivers are the objects or elements within the set, i.e., rivers of India. The river Nile is not part of rivers of India. So it is not part of this particular set. 

To define sets, we also need to learn different representations of sets. 

There are three main forms of set representations. 

1. Roster or Tabular Form

In this form or representation of a set, we simply list all the elements. Commas separate them. All the elements are within curly brackets.

Example: $\mathrm{A}=\{\mathrm{2,4},\mathrm{6,8},10\}$

Important: The order of elements does not matter, and duplicates are not written.

2. Set-builder Form

In this form, a set is described by a property that its elements satisfy.

Example: $\mathrm{A}=\{\mathrm{x}:\mathrm{x}$ is an even natural number less than 12 $\}$

This is read as ' A is the set of all x such that x is an even natural number less than 12.'

3. Venn Diagram Representation

A Venn diagram is a visual representation of sets. The universal set is usually represented by a rectangle, and subsets are represented by circles within the rectangle.

1. Empty Set or Null Set: A set with no elements. Denoted by $\}$ or $\varnothing$ . Read more about empty sets definition.

• Example: $\mathrm{A}=\{\mathrm{x}:\mathrm{x}$ is a natural number less than 1 $\}$

2. Finite Set: A set with a finite number of elements.

• Example: B = {1, 2, 3, 4, 5}

3. Infinite Set: A set with an uncountable or infinite number of elements.

• Example: $\mathrm{C}=\{\mathrm{x}:\mathrm{x}$ is a natural number $\}$

4. Equal Sets: Two sets A and B are equal if they have exactly the same elements. Learn more equal sets side-by-side. 

• Example: $A=\{\mathrm{1,2},3\},B=\{\mathrm{3,1},2\}\Rightarrow A=B$

5. Equivalent Sets: Two sets are equivalent if they have the same number of elements. It does not depend on what the elements are.

• Example: $\mathrm{A}=\{\mathrm{1,2},3\},\mathrm{B}=\{\mathrm{a},\mathrm{b},\mathrm{c}\}\Rightarrow \mathrm{A}$ and B are equivalent

6. Singleton Set: A set with only one element.

• Example: A = {0}

7. Subset: $A$ is a subset of $B$  if every element of $A$  is also in $B$ . Denoted by $A\subseteq B$ .

8. Proper Subset : If $A\subseteq B$ and $A\ne B$ , then $A$ is a proper subset of $B$ . Denoted by $A\subset B$ .

9. Power Set : The set of all subsets of a given set A. Denoted by $\mathrm{P}\left(\mathrm{A}\right)$ .

10. Universal Set : A set that contains all elements under consideration. Denoted by U. There is more to universal set that you may want to learn about. 

11. Disjoint Sets: Two sets with no common elements.

• Example: $\mathrm{A}=\left\{\mathrm{1,2}\right\},\mathrm{B}=\left\{\mathrm{3,4}\right\}\Rightarrow \mathrm{A}\cap \mathrm{B}=\varnothing$

So, what is a cardinality of a set?
The number of elements in a set is called the cardinality of the set. It's also known as the size of the set. It is denoted by $\left|\mathrm{A}\right|$  for a set A.
Example: If $\mathrm{A}=\{\mathrm{1,2},\mathrm{3,4}\}$ , then $\left|\mathrm{A}\right|=4$ .

When we talk about set notation, it is what one set is part of. In this case, they are within curly brackets { }.
We keep the elements within specific symbols. This is to define the set's elements. Or, it could be to how two sets are linked.  

$\in$ : Element of

$\notin :$ Not an element of

$\subseteq$ : Subset of

C : Proper subset of

$\not\subset :$ : Not a subset of

$\mathrm{\varnothing }$ or $\}$ : Empty set

|A| : Cardinality of set A

N : Set of natural numbers $=\{\mathrm{1,2},3,\dots \}$

W : Set of whole numbers $=\{\mathrm{0,1},\mathrm{2,3},\dots \}$

Z : Set of integers $=\{\dots ,-2,-\mathrm{1,0},\mathrm{1,2},\dots \}$

Q : Set of rational numbers

R : Set of real numbers

C : Set of complex numbers