Question

A student wants to pair up natural numbers such that they satisfy the equation $2x + y = 41$, where $x, y \in \mathbb{N}$. 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.

Hint:

To check whether a relation is an equivalence relation, verify if it is reflexive, symmetric, and transitive.

Answer 1

We are given the equation $2x + y = 41$, where $x, y \in \mathbb{N}$. Let's first find the domain and range of the relation. - Domain: Since $x$ is a natural number, for each value of $x$, we can solve for $y$ using the equation $y = 41 - 2x$. Hence, the domain is the set of natural numbers such that $41 - 2x$ is a natural number. The value of $x$ should be such that $41 - 2x>0$, which gives: \[ x<\frac{41}{2} = 20.5. \] Thus, $x \in \{1, 2, 3, \dots, 20\}$. So the domain of the relation is $\{1, 2, 3, \dots, 20\}$. - Range: From the equation $y = 41 - 2x$, we see that as $x$ ranges from 1 to 20, the corresponding values of $y$ will be the set $\{39, 37, 35, \dots, 1\}$. Therefore, the range of the relation is $\{1, 3, 5, \dots, 39\}$. - Reflexivity: A relation is reflexive if every element is related to itself. For reflexivity, we would need $2x + x = 41$, or $3x = 41$, which is not possible since 41 is not divisible by 3. Thus, the relation is not reflexive. - Symmetry: A relation is symmetric if for every pair $(x, y)$, the pair $(y, x)$ is also in the relation. However, for this relation, we do not have symmetry because if $(x, y)$ satisfies the equation, $(y, x)$ does not. Therefore, the relation is not symmetric. - Transitivity: A relation is transitive if for any pairs $(x, y)$ and $(y, z)$ in the relation, the pair $(x, z)$ also satisfies the equation. For this case, we can check that the relation does not satisfy transitivity because there is no direct connection between $x$ and $z$. Thus, the relation is not transitive. - Conclusion: Since the relation is neither reflexive, symmetric, nor transitive, it is not an equivalence relation.