CBSE 2025, Set 1

Question

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.

Edvaya · Accepted Answer

The correct answer is Subjective. Let the relation be $R$ defined on the set of natural numbers $N$. $R = \{(x, y) : 2x + y = 41, x, y \in N\}$ From the equation, $y = 41 - 2x$. Since $y \in N$, $y \ge 1$. $41 - 2x \ge 1$ $2x \le 40 \implies x \le 20$ Since $x \in N$, $x \ge 1$. Thus, $x$ can take natural values from $1$ to $20$. *1. Domain and Range:* Domain of $R = \{x : (x,y) \in R\} = \{1, 2, 3, \dots, 20\}$ Range of $R = \{y : (x,y) \in R\}$. Plugging in $x$ values into $y = 41-2x$: For $x=1, y=39$ For $x=2, y=37$ ... For $x=20, y=1$ Range $= \{39, 37, 35, \dots, 3, 1\}$ (or all odd natural numbers up to 39). *2. Equivalence Relation Checks:* *(i) Reflexive:* For $R$ to be reflexive, $(x, x) \in R$ for all $x \in N$. If $x = x$, then $2x + x = 3x$. For $(1,1)$, $3(1) = 3 
e 41$. Since $(1, 1) 
otin R$, the relation is *not reflexive*. *(ii) Symmetric:* For $R$ to be symmetric, if $(x, y) \in R$, then $(y, x) \in R$. Let $(1, 39) \in R$ (since $2(1) + 39 = 41$). Let us check $(39, 1)$: $2(39) + 1 = 78 + 1 = 79 
e 41$. Since $(39, 1) 
otin R$, the relation is *not symmetric*. *(iii) Transitive:* For $R$ to be transitive, if $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$. Consider $(19, 3) \in R$ (since $2(19) + 3 = 41$) and $(3, 35) \in R$ (since $2(3) + 35 = 41$). Let us check $(19, 35)$: $2(19) + 35 = 38 + 35 = 73 
e 41$. Since $(19, 35) 
otin R$, the relation is *not transitive*. *Conclusion:* Since the relation $R$ is neither reflexive, symmetric, nor transitive, it is *not an equivalence relation*.

Edvaya Target

Edvaya Aspire

Take a Free Mock Test

Target Subjects

Aspire Subjects

Subject Mastery

Mathematics > Sets Relations And Functions > Relations And Functions

Keep Practicing

Current Question

Community Stats

Track Your Progress

Edvaya Target

We Value Your Privacy