Let X = R × R. Define a relation R on X as: (a_1, b_1) R (a_2, b_2) ⇔ b_1 = b_2.
Statement I: R is a

Q: Why is this an equivalence relation? It feels too simple.

Mathematics is often about stripping away the noise. An equivalence relation is just a way of partitioning a set into 'buckets' of similarity. Here, the 'bucket' is defined by the $y$-coordinate. Since equality is reflexive, symmetric, and transitive, any relation based on equality is inherently an equivalence relation.

Q: Why is the set $S$ a line and not just a single point?

Great question! The set $S$ is an equivalence class. It contains *all* points that satisfy the relation with $(a, b)$. Since the relation only cares about the $y$-coordinate, any point $(x, b)$ works, regardless of what $x$ is. Thus, you get the entire horizontal line $y = b$.

Q: How do I determine if two lines are parallel?

Parallel lines must have the same slope. The line $y = b$ is a horizontal line with a slope of $0$. The line $y = x$ has a slope of $1$. Since $0 \neq 1$, they cannot be parallel; they must intersect.

Question

Let $X=\mathbb{R}	imes\mathbb{R}$. Define a relation R on X as: $(a_{1},b_{1})R(a_{2},b_{2})\Leftrightarrow b_{1}=b_{2}$.
Statement I: R is an equivalence relation.
Statement II: For some $(a,b)\in X$ the set $S=\{(x,y)\in X : (x,y)R(a,b)\}$ represents a line parallel to $y=x$.
In the light of the above statements, choose the correct answer from the options given below:

Options:

(A) Both Statement I and Statement II are false

(B) Statement I is true but Statement II is false

(C) Both Statement I and Statement II are true

(D) Statement I is false but Statement II is true

Sigma Percentile · Accepted Answer

### Analyzing the Universe of Points

Welcome, fellow traveler of the Cartesian plane. Today, we are exploring the architecture of relationships within the set $X = \mathbb{R} 	imes \mathbb{R}$.

Imagine an infinite sheet of graph paper, stretching in every direction, where every point is defined by an ordered pair $(a, b)$. This is our universe.

### The Anatomy of the Relation

We are introduced to a relation $R$ defined by $(a_1, b_1) R (a_2, b_2) \iff b_1 = b_2$. This is the heartbeat of the problem.

It tells us that two points are 'related' if they share the same horizontal level. If you are standing at $(2, 3)$ and your friend is at $(5, 3)$, you are related because you both share the same $y$-coordinate, $3$.

This is the core geometric reality: the relation $R$ is simply a way of grouping points that lie on the same horizontal line.

### The Equivalence Test

To determine if $R$ is an equivalence relation, we must test the three pillars of logic.

**Reflexivity:** Does a point $(a, b)$ relate to itself? Since $b = b$, the answer is a resounding yes.

**Symmetry:** If $(a_1, b_1) R (a_2, b_2)$, then $b_1 = b_2$. Does this imply $b_2 = b_1$? Absolutely. The relation is symmetric.

**Transitivity:** If $(a_1, b_1) R (a_2, b_2)$ and $(a_2, b_2) R (a_3, b_3)$, then $b_1 = b_2$ and $b_2 = b_3$. It follows that $b_1 = b_3$.

The relation is transitive. Because it satisfies all three, **Statement I is undeniably true.**

### The Geometric Trap

Now, we arrive at Statement II. We are asked to analyze the set $S = \{(x, y) \in X : (x, y) R (a, b)\}$.

By the definition of our relation, this set contains all points $(x, y)$ such that $y = b$. Geometrically, this is a horizontal line passing through the point $(a, b)$.

The slope of any horizontal line is $0$. Statement II claims this line is parallel to $y = x$.

However, the line $y = x$ has a slope of $1$. Since $0 
eq 1$, these lines are not parallel; they intersect at $(b, b)$.

Thus, **Statement II is false.**

### The Takeaway

We have navigated the logic of equivalence relations and the geometry of the Cartesian plane.

We proved that while the relation creates a beautiful structure of horizontal lines, those lines are distinct from the diagonal line $y = x$.

Keep this distinction in mind: equivalence relations partition space, but they do not always align with the lines we expect. Stay curious, and keep exploring the geometry behind the algebra.

Select Answer:

Visualized Solution

Understanding the Set $X$

Defining the Relation $R$

Checking Reflexivity

Checking Symmetry

Checking Transitivity

Conclusion for Statement I

Analyzing Statement II

Geometric Interpretation

Final Conclusion

The Sigma Insight: Types of Relations

Analyzing the Universe of Points

The Anatomy of the Relation

The Equivalence Test

The Geometric Trap

The Takeaway

Similar Questions

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