Class CBSE Class 12 Mathematics Relations and Functions Q #1453
COMPETENCY BASED
UNDERSTAND
3 Marks 2025 AISSCE(Board Exam) SA
A student wants to pair up natural numbers in such a way that they satisfy the equation $2x+y=41$, $x, y\in N$. Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or not.
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Detailed Solution

Step 1: Express $y$ in terms of $x$

Given the equation $2x + y = 41$, where $x, y \in N$ (natural numbers). We can express $y$ in terms of $x$ as follows: $y = 41 - 2x$

Step 2: Determine the domain

Since $x$ and $y$ are natural numbers, $x > 0$ and $y > 0$. $y = 41 - 2x > 0$ $2x < 41$ $x < \frac{41}{2}$ $x < 20.5$ Since $x$ is a natural number, the possible values of $x$ are $1, 2, 3, ..., 20$. Therefore, the domain of the relation is $\{1, 2, 3, ..., 20\}$.

Step 3: Determine the range

For each value of $x$ in the domain, we can find the corresponding value of $y$: When $x = 1$, $y = 41 - 2(1) = 39$ When $x = 2$, $y = 41 - 2(2) = 37$ When $x = 3$, $y = 41 - 2(3) = 35$ ... When $x = 20$, $y = 41 - 2(20) = 1$ The range of the relation is $\{1, 3, 5, ..., 39\}$.

Step 4: Check for reflexivity

For the relation to be reflexive, $(x, x)$ must be in the relation for all $x$ in the domain. This means $2x + x = 41$, or $3x = 41$, which gives $x = \frac{41}{3}$, which is not a natural number. Therefore, the relation is not reflexive.

Step 5: Check for symmetry

For the relation to be symmetric, if $(x, y)$ is in the relation, then $(y, x)$ must also be in the relation. If $(x, y)$ is in the relation, then $2x + y = 41$. If $(y, x)$ is in the relation, then $2y + x = 41$. Let's take an example: $x = 1$, $y = 39$. Then $2(1) + 39 = 41$. Now, check if $2(39) + 1 = 41$. $2(39) + 1 = 78 + 1 = 79 \neq 41$. Therefore, the relation is not symmetric.

Step 6: Check for transitivity

For the relation to be transitive, if $(x, y)$ and $(y, z)$ are in the relation, then $(x, z)$ must also be in the relation. If $(x, y)$ is in the relation, then $2x + y = 41$. If $(y, z)$ is in the relation, then $2y + z = 41$. We need to check if $2x + z = 41$. From the first equation, $y = 41 - 2x$. Substituting this into the second equation, $2(41 - 2x) + z = 41$, which simplifies to $82 - 4x + z = 41$, so $z = 41 + 4x - 82 = 4x - 41$. Now, we need to check if $2x + z = 41$. $2x + (4x - 41) = 41$ $6x - 41 = 41$ $6x = 82$ $x = \frac{82}{6} = \frac{41}{3}$, which is not a natural number. Let's take an example: Let $x = 1$, then $y = 41 - 2(1) = 39$. So, $(1, 39)$ is in the relation. Let $y = 39$, then $2(39) + z = 41$, so $78 + z = 41$, which means $z = 41 - 78 = -37$. But $z$ must be a natural number, so this case is not valid. Let's try another approach. Suppose $(x, y)$ and $(y, z)$ are in the relation. Then $2x + y = 41$ and $2y + z = 41$. We want to see if $2x + z = 41$. From the first equation, $y = 41 - 2x$. Substituting into the second equation, $2(41 - 2x) + z = 41$, so $82 - 4x + z = 41$, which gives $z = 4x - 41$. Now we check if $(x, z)$ is in the relation, i.e., if $2x + z = 41$. $2x + (4x - 41) = 6x - 41$. We want to see if $6x - 41 = 41$, which means $6x = 82$, so $x = \frac{41}{3}$, which is not a natural number. Therefore, the relation is not transitive.

Step 7: Check for equivalence relation

Since the relation is not reflexive, symmetric, or transitive, it is not an equivalence relation.

Final Answer: Domain: {1, 2, 3, ..., 20}, Range: {1, 3, 5, ..., 39}. The relation is not reflexive, symmetric, or transitive. Therefore, it is not an equivalence relation.

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the given equation and the properties of natural numbers to determine the domain and range, and then apply the definitions of reflexive, symmetric, and transitive relations to check if the given relation satisfies these properties.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding of the concepts of domain, range, reflexive, symmetric, and transitive relations, and how these concepts apply to a specific equation.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question requires the student to apply their knowledge of relations and functions to a specific problem and to analyze the properties of the relation.

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