Why is it reflexive? I thought reflexive meant $x$ relates to $y$?

Reflexivity is about the self. It asks: 'Is $x$ in the same room as $x$?' Since $x$ is always in the same room as itself, it is reflexive. It is a fundamental property of identity.

What if $A_i$ is empty? Does the logic still hold?

The definition of a partition usually implies non-empty subsets. However, even if a subset were empty, the logic holds vacuously because there would be no elements $x$ in that subset to violate the conditions.

Is every equivalence relation a partition?

Yes! This is the Fundamental Theorem of Equivalence Relations. Every equivalence relation partitions the set into disjoint equivalence classes. It is a beautiful, two-way street in mathematics.

JEE Main 2022 (29 June Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Sets and Relations: Let a set $A = A_{1} \cup A_{2} \cup \dots \cup A_{k}$ , where $A_{i} \cap A_{j} = ϕ$ for $i \neq = j, 1 \leq i, j \leq k$ . Define the relation $R$ from $A$ to $A$ by $R = {(x, y) : y \in A_{i} if and only if x \in A_{i}, 1 \leq i \leq k}$ . Then, $R$ is:

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Visualized Solution

Visualizing the Partition of Set $A$

Set $A$ is partitioned into $k$ disjoint subsets: $A_{1}, A_{2}, \dots, A_{k}$ .
Disjoint means $A_{i} \cap A_{j} = ϕ$ for all $i \neq = j$ .

Understanding the Relation $R$

Relation $R$ is defined as: $(x, y) \in R ⟺ x, y \in A_{i}$ for some $i$ .
Elements are related if they belong to the same subset.

Defining Reflexivity

Reflexivity Check: A relation $R$ is reflexive if $(x, x) \in R$ for all $x \in A$ .

Applying Reflexivity Logic

For any $x \in A$ , there exists exactly one $i$ such that $x \in A_{i}$ .
Clearly, $x \in A_{i} ⟺ x \in A_{i}$ is a tautology.

Reflexivity Confirmed

Therefore, $(x, x) \in R$ for all $x \in A$ .
The relation $R$ is reflexive.

Defining Symmetry

Symmetry Check: A relation $R$ is symmetric if $(x, y) \in R ⟹ (y, x) \in R$ .

Applying Symmetry Logic

Assume $(x, y) \in R$ . This means $x, y \in A_{i}$ for some $i$ .
This is equivalent to saying $y, x \in A_{i}$ .

Symmetry Confirmed

Thus, $(y, x) \in R$ .
The relation $R$ is symmetric.

Defining Transitivity

Transitivity Check: A relation $R$ is transitive if $(x, y) \in R$ and $(y, z) \in R ⟹ (x, z) \in R$ .

Applying Transitivity Logic

Assume $(x, y) \in R$ and $(y, z) \in R$ .
$(x, y) \in R ⟹ x, y \in A_{i}$ for some $i$ .
$(y, z) \in R ⟹ y, z \in A_{j}$ for some $j$ .
Since $y \in A_{i}$ and $y \in A_{j}$ , and subsets are disjoint, $i$ must equal $j$ .

Transitivity Confirmed

Thus, $x, z \in A_{i}$ , which means $(x, z) \in R$ .
The relation $R$ is transitive.

The Final Verdict: Equivalence Relation

Since $R$ is reflexive, symmetric, and transitive, it is an equivalence relation.
Key Takeaway: Every partition of a set defines an equivalence relation on that set.

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The Sigma Insight: Equivalence Relations

Solution Diagram

Analyzing the Setup

Consider a set

A

that is partitioned into

k

distinct, non-overlapping subsets (rooms) denoted by

A_{1}, A_{2}, \dots, A_{k}

The relation

R

is defined such that two elements

x

and

y

are related if and only if they reside in the same room

A_{i}

. This structure serves as the foundation for understanding equivalence relations.

Testing the Three Pillars

To confirm that

R

is an equivalence relation, we must verify the three fundamental properties: Reflexivity, Symmetry, and Transitivity.

1. Reflexivity: Does

x

relate to

x

? Since

x

is an element of some room

A_{i}

, it is trivially in the same room as itself. Thus,

(x, x) \in R

holds for all

x \in A

2. Symmetry: If

x

is in the same room as

y

, it follows by definition that

y

is in the same room as

x

. Therefore, the implication

(x, y) \in R ⟹ (y, x) \in R

is satisfied.

3. Transitivity: Suppose

(x, y) \in R

and

(y, z) \in R

. This implies that

x

and

y

are in the same room

A_{i}

, and

y

and

z

are in the same room

A_{i}

Consequently,

x, y,

and

z

must all reside within the same room

A_{i}

. This confirms that

(x, z) \in R

Conclusion

Because the relation

R

satisfies reflexivity, symmetry, and transitivity, we conclude that $R$ is an equivalence relation.

This result demonstrates the Fundamental Theorem of Equivalence Relations, which establishes a one-to-one correspondence between the partitions of a set and the equivalence relations defined upon it.

Select Answer:

Visualized Solution

Visualizing the Partition of Set $A$

Understanding the Relation $R$

Defining Reflexivity

Applying Reflexivity Logic

Reflexivity Confirmed

Defining Symmetry

Applying Symmetry Logic

Symmetry Confirmed

Defining Transitivity

Applying Transitivity Logic

Transitivity Confirmed

The Final Verdict: Equivalence Relation

The Sigma Insight: Equivalence Relations

Analyzing the Setup

Testing the Three Pillars

Conclusion

Similar Questions

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Select Answer:

Visualized Solution

Visualizing the Partition of Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

Understanding the Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

Defining Reflexivity

Applying Reflexivity Logic

Reflexivity Confirmed

Defining Symmetry

Applying Symmetry Logic

Symmetry Confirmed

Defining Transitivity

Applying Transitivity Logic

Transitivity Confirmed

The Final Verdict: Equivalence Relation

The Sigma Insight: Equivalence Relations

Analyzing the Setup

Testing the Three Pillars

Conclusion

Similar Questions

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Visualizing the Partition of Set $A$

Understanding the Relation $R$

Let $R$ be a relation defined on $N$ as $a R b$ is $2 a + 3 b$ is a multiple of 5, $a, b \in N$ . Then $R$ is

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Define a relation R over a class of $n \times n$ real matrices A and B as "ARB iff there exists a non-singular matrix P such that $P A P^{- 1} = B$ ". Then which of the following is true?

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The minimum number of elements that must be added to the relation $R = {(a, b), (b, c), (b, d)}$ on the set ${a, b, c, d}$ so that it is an equivalence relation, is ______.