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

The given equation is $2x + y = 41$, where $x, y \in \mathbb{N}$. We analyze the properties of this relation: - Domain: Since $y = 41 - 2x$ must be a natural number, $41 - 2x>0$. This implies $x<41/2 = 20.5$. Therefore, $x \in \{1, 2, \dots, 20\}$. The domain is $\{1, 2, \dots, 20\}$. - Range: For $x \in \{1, 2, \dots, 20\}$, the corresponding $y$ values are $\{39, 37, \dots, 1\}$. The range is $\{1, 3, \dots, 39\}$. - Reflexivity: For a relation to be reflexive, $2x + x = 41$, or $3x = 41$. This equation has no integer solution for $x$, so the relation is not reflexive. - Symmetry: If $(x, y)$ is in the relation, then $(y, x)$ must also be. For this relation, if $2x + y = 41$, it does not necessarily mean $2y + x = 41$. Thus, the relation is not symmetric. - Transitivity: If $(x, y)$ and $(y, z)$ are in the relation, then $(x, z)$ must also be. With $2x + y = 41$ and $2y + z = 41$, there is no direct condition guaranteeing $2x + z = 41$. Therefore, the relation is not transitive. - Conclusion: As the relation is not reflexive, symmetric, or transitive, it is not an equivalence relation.