Let the relations R1 and R2 on the set X = {1, 2, 3, ..., 20} be given by R1 = {(x, y) : 2x - 3y = 2

Q: Wait, why do we only check for $y$ values up to 12 for $R_1$? What if $y$ is larger?

Great observation! We are constrained by the set $X = \{1, 2, \dots, 20\}$. If $y$ were larger than 12, say $y=14$, then $x = \frac{3(14)+2}{2} = 22$, which is outside our set $X$. We must strictly adhere to the domain and codomain defined by $X$.

Q: Do I need to worry about the identity relation $(x, x)$?

Excellent question. A relation is symmetric if $(x, y) \in R \implies (y, x) \in R$. If a pair is of the form $(x, x)$, its reverse is itself, so it is already symmetric. In this problem, none of our pairs satisfy $x=y$, so we don't have to worry about them, but always keep that in mind!

Q: Why can't we just add the reverse pairs and be done? Is there a case where adding one pair makes another pair symmetric automatically?

In this specific problem, the pairs are distinct and do not overlap in a way that creates a chain. However, you are thinking like a mathematician! If we had a pair $(a, b)$ and we added $(b, a)$, we would only need to add more if $(b, a)$ was not already there. Since our sets $R_1$ and $R_2$ contain no reflexive or pre-existing symmetric pairs, each pair requires exactly one unique addition.

Q: Is it possible for $M$ or $N$ to be zero?

Yes, absolutely! If a relation were already symmetric, $M$ or $N$ would be zero. For example, if $R$ contained $(2, 3)$ and $(3, 2)$, no additions would be needed for those specific elements. Always check if the set is already symmetric before counting.

Q: Why is the answer 10? Did I miss any pairs?

Let's re-verify: $R_1$ has 6 pairs, none of which are symmetric, so $M=6$. $R_2$ has 4 pairs, none of which are symmetric, so $N=4$. $6+4=10$. You haven't missed anything; the logic holds firm because the sets are disjoint and contain no self-symmetric elements.

Question

Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, \dots, 20\}$ be given by $R_1 = \{(x, y) : 2x - 3y = 2\}$ and $R_2 = \{(x, y) : -5x + 4y = 0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M + N$ equals

Options:

(A) (1) 12

(B) (2) 16

(C) (3) 8

(D) (4) 10

Sigma Percentile · Accepted Answer

### The Geometry of Symmetry: A Journey into Relations

Welcome, fellow traveler of the mathematical landscape. Today, we are going to peel back the layers of a problem that, at first glance, seems like a simple exercise in counting.

But beneath the surface of these algebraic relations lies a beautiful geometric truth: the concept of symmetry.

Imagine the set $X = \{1, 2, 3, \dots, 20\}$ as a vast, discrete grid. Every relation we define on this set is essentially a collection of points on this grid.

When we talk about a relation being 'symmetric,' we are talking about a mirror. If you place a mirror along the line $y = x$, every point $(x, y)$ in your relation must have a corresponding reflection $(y, x)$ on the other side.

If that reflection is missing, the relation is 'broken'—it lacks symmetry. Our job today is to fix these broken relations by adding the minimum number of points required to complete the reflection.

### Unmasking $R_1$: The Parity Puzzle

Let us look at $R_1 = \{(x, y) : 2x - 3y = 2\}$. This is not just an equation; it is a constraint on the existence of points.

We can rewrite this as $2x = 3y + 2$. Immediately, our mathematical intuition should fire: $2x$ is always even. Therefore, $3y + 2$ must also be even.

For $3y + 2$ to be even, $3y$ must be even, which forces $y$ to be an even number. We test our values of $y$ within the set $X$:

$$2x = 3y + 2$$

- If $y=2$, $2x = 8 \implies x=4$. We have the point $(4, 2)$.
- If $y=4$, $2x = 14 \implies x=7$. We have the point $(7, 4)$.
- If $y=6$, $2x = 20 \implies x=10$. We have the point $(10, 6)$.
- If $y=8$, $2x = 26 \implies x=13$. We have the point $(13, 8)$.
- If $y=10$, $2x = 32 \implies x=16$. We have the point $(16, 10)$.
- If $y=12$, $2x = 38 \implies x=19$. We have the point $(19, 12)$.

We have found 6 distinct points. To make $R_1$ symmetric, we need the reflection of each of these points.

Since none of the points $(2, 4), (4, 7), (6, 10), (8, 13), (10, 16),$ or $(12, 19)$ are currently in our set, we must add all 6 of them. Thus, **$M = 6$**.

### The Elegance of $R_2$: Multiples and Ratios

Now, let us turn our attention to $R_2 = \{(x, y) : -5x + 4y = 0\}$. This rearranges beautifully into the following form:

$$y = \frac{5}{4}x$$

This tells us that for $y$ to be an integer, $x$ must be a multiple of 4. Let's walk through the possibilities:

- For $x=4$, $y=5$, giving us $(4, 5)$.
- For $x=8$, $y=10$, giving us $(8, 10)$.
- For $x=12$, $y=15$, giving us $(12, 15)$.
- For $x=16$, $y=20$, giving us $(16, 20)$.

If we try $x=20$, $y=25$, which is outside our set $X$. We have 4 points: $(4, 5), (8, 10), (12, 15), (16, 20)$.

Just like before, none of these points are their own reflections, and none of their reflections exist in the set. To restore symmetry, we must add the 4 reverse pairs: $(5, 4), (10, 8), (15, 12),$ and $(20, 16)$.

Therefore, **$N = 4$**.

### The Final Synthesis

We have navigated the constraints, identified the points, and recognized the missing reflections. The total number of elements to be added is simply the sum of our requirements for both relations:

$$M + N = 6 + 4 = 10$$

It is a simple result, but look at the journey! We used parity to solve $R_1$ and divisibility to solve $R_2$.

We treated the relations not as abstract symbols, but as physical entities on a grid. Keep this perspective, and you will find that even the most complex JEE problems are just stories waiting to be told.

Select Answer:

Visualized Solution

Problem Setup

Concept of Symmetric Relations

Analyzing Relation $R_{1}$

Finding Elements of $R_{1}$

Checking Symmetry in $R_{1}$

Determining $M$

Analyzing Relation $R_{2}$

Finding Elements of $R_{2}$

Checking Symmetry in $R_{2}$

Determining $N$

Calculating $M + N$

The Sigma Insight: Types of Relations

The Geometry of Symmetry

Unmasking $R_{1}$

The Elegance of $R_{2}$

The Final Synthesis

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Select Answer:

Visualized Solution

Problem Setup

Concept of Symmetric Relations

Analyzing Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R1​

Finding Elements of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R1​

Checking Symmetry in .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R1​

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