Relations

Given: Relation ρ on natural numbers: $%CF%81%20%3D%20%5C%7B(x%2C%20y)%20%5Cin%20%5Cmathbb%7BN%7D%20%5Ctimes%20%5Cmathbb%7BN%7D%20%3A%202x%20%2B%20y%20%3D%204%5C%7D$

Step 1: We are looking for natural number solutions to the equation: $2x%20%2B%20y%20%3D%204$
Rewriting: $y%20%3D%204%20-%202x$

Step 2: Try natural number values of $x$ :

For $x%20%3D%201$ : $y%20%3D%204%20-%202(1)%20%3D%202$ → valid (since $y%20%5Cin%20%5Cmathbb%7BN%7D$ )
For $x%20%3D%202$ : $y%20%3D%204%20-%204%20%3D%200$ → not valid (0 ∉ ℕ)
For $x%20%3D%203$ : $y%20%3D%204%20-%206%20%3D%20-2$ → not valid

So, only valid ordered pair is $(1%2C%202)$

Step 3: Now, find domain A and range B:

Domain A: set of first elements = $%5C%7B1%5C%7D$
Range B: set of second elements = $%5C%7B2%5C%7D$

These sets are subsets of:

$A%20%5Csubset%20%5C%7Bx%20%5Cin%20%5Cmathbb%7BN%7D%20%3A%201%20%5Cle%20x%20%5Cle%2020%5C%7D$
$B%20%5Csubset%20%5C%7By%20%5Cin%20%5Cmathbb%7BN%7D%20%3A%201%20%5Cle%20y%20%5Cle%2039%5C%7D$

Final Answer: (A): $A%20%5Csubset%20%5C%7Bx%20%5Cin%20%5Cmathbb%7BN%7D%20%3A%201%20%5Cle%20x%20%5Cle%2020%5C%7D%2C%20B%20%5Csubset%20%5C%7By%20%5Cin%20%5Cmathbb%7BN%7D%20%3A%201%20%5Cle%20y%20%5Cle%2039%5C%7D$

Transitivity means that for a relation $R$ , if $(a%2C%20b)%20%5Cin%20R$ and $(b%2C%20c)%20%5Cin%20R$ , then $(a%2C%20c)%20%5Cin%20R$ must also hold.

We are asked whether the composition of two transitive relations is necessarily transitive.

Let’s take a counterexample to demonstrate that it may not be transitive.

Let the set $X%20%3D%20%5C%7B1%2C%202%2C%203%5C%7D$ , and define two relations:

$%5Crho_1%20%3D%20%5C%7B(1%2C%202)%2C%20(2%2C%203)%2C%20(1%2C%203)%5C%7D$
$%5Crho_2%20%3D%20%5C%7B(2%2C%203)%2C%20(3%2C%201)%2C%20(2%2C%201)%5C%7D$

Each of these is transitive on its own:

For $%5Crho_1$ : Since $(1%2C%202)$ and $(2%2C%203)$ are in $%5Crho_1$ , and $(1%2C%203)$ is also in $%5Crho_1$ , it satisfies transitivity.
For $%5Crho_2$ : Since $(2%2C%203)$ and $(3%2C%201)$ are in $%5Crho_2$ , and $(2%2C%201)$ is also in $%5Crho_2$ , it satisfies transitivity.

Now let’s consider the composition $%5Crho_1%20%5Ccirc%20%5Crho_2$ :

In a composition $%5Crho_1%20%5Ccirc%20%5Crho_2$ , we include $(a%2C%20c)$ if there exists a $b$ such that $(a%2C%20b)%20%5Cin%20%5Crho_2$ and $(b%2C%20c)%20%5Cin%20%5Crho_1$ .

Now check for key pairs:

$(1%2C%203)%20%5Cin%20%5Crho_1%20%5Ccirc%20%5Crho_2$ and $(3%2C%201)%20%5Cin%20%5Crho_1%20%5Ccirc%20%5Crho_2$
But $(1%2C%201)%20%5Cnotin%20%5Crho_1%20%5Ccirc%20%5Crho_2$ , so the composition is not transitive.

This violates transitivity: since $(1%2C%203)$ and $(3%2C%201)$ are in the relation, then $(1%2C%201)$ should also be in the relation, but it is not.

Therefore, the composition $%5Crho_1%20%5Ccirc%20%5Crho_2$ is not necessarily transitive.

Correct option: (B) $%5Crho_1%20%5Ccirc%20%5Crho_2$ is not a transitive relation

About Relations - WBJEE

Relations is a vital chapter for WBJEE aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Relations PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Relations carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

About Relations - WBJEE

Frequently Asked Questions

Why focus on Relations PYQs?

How to best use this analysis?

Relations

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

About Relations - WBJEE

Frequently Asked Questions

Why focus on Relations PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs