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

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Relations

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs