Equal Sets: Class 11 Maths Notes, Definition, Representation, Properties & Solved Examples

Equal sets are sets that contain exactly the same elements. Importantly, the order in which elements are listed in a set does not matter.

So, a basic equal set example is

$\bar{A}=\{\mathrm{1,2},\mathrm{3,4},5\})$  and $B=\{\mathrm{5,4},\mathrm{3,2},1\}$

They are equal sets, as they have the same elements, regardless of the order that is different.

Importance of Equal Sets (for Engineering Exams)

• The JEE Mains syllabus focuses on core concepts of set theory and the types of sets, such as equal sets. Learning what is an equal set can be of great help, where you can easily secure some good marks.  
• The questions on JEE focusing on domains, relations, or ranges, focus on testing students whether they can recognise equal sets.

The NCERT definition of equal sets is in the 1.5 section of the textbook. It says,

"Two sets A and B are said to be equal if they have exactly the same elements and we write A = B."

The basic explanation of equal sets goes like this. 

Mathematically, this is denoted by $A=B$ .

Example
Let $A=\{\mathrm{1,2},\mathrm{3,4},5\}$  and $B=\{\mathrm{1,2},\mathrm{3,4},5\}$ . Here:

• Every element of $A$ is in $B$ .
• Every element of B is in A.

Thus, $\mathrm{A}=\mathrm{B}$ .

 

Unequal Sets

Two sets are called unequal if they do not have exactly the same elements. Even if one element differs, the sets are not equal.

Example

$\begin{array}{rr}& -A=\{\mathrm{1,2},\mathrm{3,4},5\}\\ & -D=\{\mathrm{2,5},6\}\end{array}$

Here, sets A and D have different elements and different numbers of elements. Thus, $\mathrm{A}\ne \mathrm{D}$ .

The concept of equivalent sets are a little different from equal sets.

To define equivalent sets, just  know that they are sets that have the same number of elements, irrespective of whether the elements themselves are identical. 

Example

$\begin{array}{rr}& -\mathrm{A}=\{\mathrm{1,2},\mathrm{3,4},5\}\\ & -\mathrm{C}=\{\mathrm{2,4},\mathrm{6,7},9\}\end{array}$

Here, A and C both have five elements. Hence, they are equivalent sets, even though their elements differ.

The symbol for equivalent sets is $\sim$ or $\equiv$ .

We can visually represent equal sets with Venn diagrams.

In a Venn diagram, if two sets $A$  and $B$ are equal, their diagrams will perfectly overlap. This shows that they share exactly the same elements.

For example, if $A=\left\{\mathrm{11,17,38}\right\}$  and $B=\left\{\mathrm{11,17,38}\right\}$ , then the Venn diagram would show two overlapping circles with elements 11,17 , and 38 inside.

Key properties of equal sets include the following. 

1. Order of Elements

• The order of elements in a set does not affect its equality with another set.

2. Cardinality

• Equal sets have the same number of elements (same cardinality).

3. Subset Relationship

• If $A\subseteq B$ and $B\subseteq A$ , then $A=B$ .

4. Equality of Power Sets

• The power sets (the set of all subsets) of equal sets have the same number of elements.

5. Relation to Equivalent Sets

• All equal sets are equivalent sets because they have the same cardinality.
• However, not all equivalent sets are equal.

Explore the main differences between equal sets and equivalent sets below.  

Aspect	Equal Sets	Equivalent Sets
Definition	Two sets are equal with exactly the same elements.	Two sets are equivalent if they have the same number of elements.
Cardinality (Size)	Same elements → same cardinality.	Same number of elements → same cardinality.
Element Matching	Every element in one set is also in the other.	Elements don’t need to match, only the quantity matters.
Notation Used	A=B	A∼B or simply compare their sizes.
Relationship	All equal sets are also equivalent.	Equivalent sets are not always equal.

1. Equal Sets: Same elements and same number of elements.

2. Equivalent Sets: Same number of elements, not necessarily the same elements.

3. The order of elements in a set is irrelevant.

4. If two sets are subsets of each other, they are equal.

5. Equal sets have the same power set.