JEE Main 2025 (January)

LEVELJEE Main

Animated Solution for Mathematics - Sets and Relations: Let $R = {(1, 2), (2, 3), (3, 3)}$ be a relation defined on the set ${1, 2, 3, 4}$ . Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Select Answer:

Progressive Hints

Hint 1

Visualize the relation

R

as a directed graph where elements of the set

A = {1, 2, 3, 4}

are nodes. An equivalence relation is essentially a collection of disjoint 'cliques' or clusters. Your first task is to ensure every node is part of a cluster by satisfying the reflexive property.

Hint 2

Remember the hierarchy of operations. Reflexivity is the 'base' requirement. Symmetry and Transitivity are 'relational' requirements. When you add an edge to satisfy transitivity, you must immediately check if symmetry is still intact. It is a dynamic process, not a static one.

Hint 3

Start by listing the required reflexive pairs:

(1, 1), (2, 2), (3, 3), (4, 4)

. Since

(3, 3)

is already present, how many more do you need? Then, look at the existing edges

(1, 2)

and

(2, 3)

and apply the definitions of symmetry and transitivity sequentially to find the remaining pairs.

Visualized Solution

The Initial Setup

Set $A = {1, 2, 3, 4}$
Initial Relation $R = {(1, 2), (2, 3), (3, 3)}$
Visualizing relations as directed graphs.

The Goal: Equivalence Relation

We need to make $R$ an Equivalence Relation.
Three conditions must be satisfied:
1. Reflexive
2. Symmetric
3. Transitive

Condition 1: Reflexivity

Reflexive Property: For every $a \in A$ , $(a, a) \in R$ .
Every node must have a self-loop.
We check elements: $1, 2, 3, 4$ .

Adding Reflexive Elements

Currently, only $(3, 3)$ is in $R$ .
Missing pairs: $(1, 1), (2, 2), (4, 4)$ .
Elements added: $3$ .

Condition 2: Symmetry

Symmetric Property: If $(a, b) \in R$ , then $(b, a) \in R$ .
For every directed arrow, there must be a reverse arrow.

Adding Symmetric Elements

We have $(1, 2) \in R ⟹$ must add $(2, 1)$ .
We have $(2, 3) \in R ⟹$ must add $(3, 2)$ .
Elements added: $2$ .

Condition 3: Transitivity

Transitive Property: If $(a, b) \in R$ and $(b, c) \in R$ , then $(a, c) \in R$ .
If there is a path of length 2, there must be a direct shortcut.

Adding Transitive Elements

We have $(1, 2)$ and $(2, 3)$ in $R$ .
Therefore, we must add the direct link $(1, 3)$ .
Elements added: $1$ .

The Ripple Effect: Re-checking Symmetry

We just added $(1, 3)$ .
To maintain Symmetry, we must immediately add its reverse: $(3, 1)$ .
Elements added: $1$ .

Final Transitivity Check

Check new pairs: $(3, 1)$ and $(1, 2) ⟹ (3, 2)$ (Already present).
$(2, 1)$ and $(1, 3) ⟹ (2, 3)$ (Already present).
$(1, 3)$ and $(3, 1) ⟹ (1, 1)$ (Already present).
No more elements needed!

Calculating the Minimum Elements Added

Reflexive additions: $(1, 1), (2, 2), (4, 4) \to 3$ elements.
Symmetric additions: $(2, 1), (3, 2) \to 2$ elements.
Transitive & Symmetric additions: $(1, 3), (3, 1) \to 2$ elements.
Total Minimum Elements Added $= 3 + 2 + 2 = 7$ .

00:00 / 00:00

The Sigma Insight: Equivalence Relations

Solution Diagram

Analyzing the Setup

We are given a set

A = {1, 2, 3, 4}

and an initial relation

R = {(1, 2), (2, 3), (3, 3)}

. To transform this into an Equivalence Relation, we must satisfy the three fundamental pillars: Reflexivity, Symmetry, and Transitivity.

The Foundation of Reflexivity

For a relation to be reflexive, every element

a \in A

must satisfy

(a, a) \in R

. Currently, only

(3, 3)

is present.

To satisfy this property for the entire set

A

, we must add the following pairs:

(1, 1), (2, 2), (4, 4)

This requires the addition of

3

elements to our relation.

The Two-Way Street of Symmetry

Symmetry dictates that if

(a, b) \in R

, then

(b, a)

must also be in

R

. We examine our current set of pairs and identify the necessary additions:

1. Since

(1, 2) \in R

, we must add

(2, 1)

. 2. Since

(2, 3) \in R

, we must add

(3, 2)

This step adds

2

elements to the relation.

The Logic of Transitivity

Transitivity requires that if

(a, b) \in R

and

(b, c) \in R

, then

(a, c)

must also be in

R

. We observe that we have

(1, 2)

and

(2, 3)

in our set.

To satisfy transitivity, we are forced to add the direct link:

(1, 3)

This adds

1

element to the relation.

The Ripple Effect

In logic, adding a new pair often triggers a chain reaction. By adding

(1, 3)

, we have violated the symmetry requirement.

To maintain symmetry, we must add the reverse of our new link:

(3, 1)

This adds

1

final element. We now perform a final sanity check: all combinations, such as

(3, 1)

and

(1, 2)

implying

(3, 2)

, are accounted for.

Final Calculation

The total number of elements added to the original relation

R

is:

3 (reflexive) + 2 (symmetric) + 1 (transitive) + 1 (symmetric correction) = 7

We have successfully constructed the Equivalence Relation by adding exactly 7 elements.

Common Traps & FAQs

Why do we need to add

(4, 4)

? It is not connected to any other element in the relation!

This is a classic trap! The definition of reflexivity is absolute: for a relation to be reflexive on set

A

, every element

a \in A

must satisfy

(a, a) \in R

. Even if

4

is an isolated island with no other connections, it must have a self-loop to be part of an equivalence relation. Think of it as a 'self-identity' requirement that cannot be skipped.

I added

(1, 3)

to satisfy transitivity, but why did I have to add

(3, 1)

? Isn't that just extra work?

It is not extra work; it is a logical necessity. If you add

(1, 3)

to satisfy transitivity, the relation is no longer symmetric unless

(3, 1)

is also present. Equivalence relations are rigid, interconnected structures; you cannot satisfy one property by breaking another. Every time you add a link, you must check if the symmetry and transitivity 'balance' is maintained.

Is there a shortcut formula to find the number of elements, or do I always have to do this manually?

While there are formulas for the total number of equivalence relations (related to Bell numbers), finding the *minimum* elements to add is best done by constructing the 'closure' of the relation. Think of it as 'completing the graph' rather than memorizing a formula. By building it step-by-step, you ensure you don't miss the subtle ripple effects that occur when you add new edges.

Animated Solution for Mathematics - Sets and Relations: Let $R = {(1, 2), (2, 3), (3, 3)}$ be a relation defined on the set ${1, 2, 3, 4}$ . Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Select Answer:

Visualized Solution

The Initial Setup

The Goal: Equivalence Relation

Condition 1: Reflexivity

Adding Reflexive Elements

Condition 2: Symmetry

Adding Symmetric Elements

Condition 3: Transitivity

Adding Transitive Elements

The Ripple Effect: Re-checking Symmetry

Final Transitivity Check

Calculating the Minimum Elements Added

The Sigma Insight: Equivalence Relations

Analyzing the Setup

The Foundation of Reflexivity

The Two-Way Street of Symmetry

The Logic of Transitivity

The Ripple Effect

Final Calculation

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Select Answer:

Visualized Solution

The Initial Setup

The Goal: Equivalence Relation

Condition 1: Reflexivity

Adding Reflexive Elements

Condition 2: Symmetry

Adding Symmetric Elements

Condition 3: Transitivity

Adding Transitive Elements

The Ripple Effect: Re-checking Symmetry

Final Transitivity Check

Calculating the Minimum Elements Added

The Sigma Insight: Equivalence Relations

Analyzing the Setup

The Foundation of Reflexivity

The Two-Way Street of Symmetry

The Logic of Transitivity

The Ripple Effect

Final Calculation

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