Let R be the set of real numbers. Statement-1: A = {(x, y) ∈ R × R : y - x is an integer} is an equi

Q: Wait, why do we check $(0, 1)$ specifically? Is there a rule for picking counter-examples?

Great question! When checking for symmetry, if you suspect it might fail, look for 'boundary' elements—specifically zero, identity, or undefined values. Zero is the classic 'symmetry breaker' in multiplication-based relations. Always test the edges of the domain!

Q: Is Relation $B$ transitive? I feel like it might be.

You have a sharp intuition! Yes, $B$ is actually transitive. If $x = \alpha y$ and $y = \beta z$, then $x = (\alpha \beta) z$. Since the product of two rationals is rational, $B$ is transitive. However, in logic, if one pillar (symmetry) fails, the whole structure is disqualified. We don't need to check transitivity to reject it!

Q: Why is $0$ considered an integer in Statement-1?

The set of integers $\mathbb{Z}$ is defined as $\{..., -2, -1, 0, 1, 2, ...\}$. Zero is absolutely an integer. Since $x - x = 0$, and $0 \in \mathbb{Z}$, the relation satisfies reflexivity perfectly. Never doubt the basics!

Question

Let R be the set of real numbers. 
Statement-1: $A = \{(x, y) \in R 	imes R : y - x 	ext{ is an integer}\}$ is an equivalence relation on R. 
Statement-2: $B = \{(x, y) \in R 	imes R : x = \alpha y 	ext{ for some rational number } \alpha\}$ is an equivalence relation on R.

Options:

(A) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B) Statement-1 is true, Statement-2 is false.

(C) Statement-1 is false, Statement-2 is true.

(D) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Sigma Percentile · Accepted Answer

### Analyzing the Three Pillars of Equivalence

In the JEE Advanced syllabus, Relations and Functions form the language of mathematical structure. To define an **Equivalence Relation**, a relation must satisfy three fundamental pillars: **Reflexivity**, **Symmetry**, and **Transitivity**.

If a relation holds these three properties, it partitions the set into disjoint equivalence classes. If it fails even one, the structure collapses.

### Statement-1: The Integer Difference

We are given the relation $A = \{(x, y) \in \mathbb{R} 	imes \mathbb{R} : y - x \in \mathbb{Z}\}$. This relation implies that two numbers are related if their difference is an integer.

**Reflexivity:** We check if $x$ is related to itself. Since $x - x = 0$ and $0$ is an integer, the relation is reflexive.

**Symmetry:** If $(x, y) \in A$, then $y - x = k$ for some $k \in \mathbb{Z}$. Consequently, $x - y = -k$. Since the negative of an integer is also an integer, $(y, x) \in A$. The relation is symmetric.

**Transitivity:** Suppose $(x, y) \in A$ and $(y, z) \in A$. Then $y - x = k_1$ and $z - y = k_2$ for some $k_1, k_2 \in \mathbb{Z}$. Adding these equations yields:
$$z - x = (z - y) + (y - x) = k_1 + k_2$$
Since the sum of two integers is an integer, $(x, z) \in A$. Thus, **Statement-1 is a valid equivalence relation.**

### Statement-2: The Rational Multiplier Trap

We examine the relation $B = \{(x, y) \in \mathbb{R} 	imes \mathbb{R} : x = \alpha y, \alpha \in \mathbb{Q}\}$.

**Reflexivity:** For any $x \in \mathbb{R}$, we can write $x = 1 \cdot x$. Since $1 \in \mathbb{Q}$, the relation is reflexive.

**Symmetry:** For symmetry, if $(x, y) \in B$, then $(y, x)$ must also be in $B$. Consider the pair $(0, 1)$. Since $0 = 0 \cdot 1$ and $0 \in \mathbb{Q}$, the pair $(0, 1)$ is in $B$.

However, for $(1, 0)$ to be in $B$, there must exist a rational $\beta$ such that:
$$1 = \beta \cdot 0$$
Because $\beta \cdot 0 = 0$ for any $\beta \in \mathbb{Q}$, it is impossible to satisfy $1 = 0$. The symmetry pillar has shattered.

### The Final Verdict

We have evaluated both statements through the lens of formal logic. Statement-1 satisfies all three pillars of an equivalence relation.

Statement-2 fails the symmetry test specifically due to the behavior of the element zero. In JEE examinations, always remember: when dealing with multiplication or division, **immediately test the zero case**, as it is the most frequent point of failure in logical relations.

Select Answer:

Visualized Solution

The Three Pillars of Equivalence

Statement 1: Reflexivity Check

Statement 1: Symmetry Check

Statement 1: Transitivity Check

Statement 1 is True

Statement 2: Reflexivity Check

Statement 2: The Symmetry Trap

Statement 2: Symmetry Failure

Final Verdict

The Sigma Insight: Types of Relations

Analyzing the Three Pillars of Equivalence

Statement-1

Statement-2

The Final Verdict

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