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Introduction

This article explains the properties of the relation $R=\{(m,n):m-n\text{ is divisible by }2\}$ on the set of natural numbers N and determines whether it is reflexive, symmetric, and transitive. Similar relations of the form “$a-b$ is divisible by $n$” are standard examples when studying equivalence relations in discrete mathematics. Understanding this structure helps in mastering questions commonly asked in competitive mathematics and computer science exams.

The question is:

Let N be the set of all natural numbers. Consider the relation R on N given by
$R=\{(m,n):m-n\text{ is divisible by }2\}$. Then
(A) R is symmetric and transitive
(B) R is symmetric but NOT transitive
(C) R is reflexive but NOT symmetric
(D) R is reflexive and transitive

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Step 1: Rewrite the relation

“$m-n$ is divisible by 2” means $m-n=2k$ for some integer $k$, so $m$ and $n$ have the same parity (both even or both odd).

Thus, on $\mathbb{N}$:

• $mRn$ ⇔ $m$ and $n$ are both even or both odd.

This is exactly the “same parity” relation.

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Check reflexive, symmetric, transitive

Reflexive property

A relation R on N is reflexive if $(n,n)\in R$ for every $n\in N$.

• For any natural number $n$, $n-n=0$.

• 0 is divisible by 2, since $0=2\cdot 0$.

• Therefore $(n,n)\in R$ for all $n\in N$, so R is reflexive.

Hence, any option claiming “not reflexive” is automatically wrong.

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Symmetric property

A relation R is symmetric if whenever $(m,n)\in R$, then $(n,m)\in R$.

• Suppose $(m,n)\in R$. Then $m-n$ is divisible by 2, so $m-n=2k$ for some integer $k$.

• Then $n-m=-(m-n)=-2k=2(-k)$, which is also divisible by 2.

• Hence $(n,m)\in R$.

So R is symmetric.

Therefore, any option that says “NOT symmetric” is incorrect.

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Transitive property

A relation R is transitive if whenever $(m,n)\in R$ and $(n,p)\in R$, then $(m,p)\in R$.

• Assume $(m,n)\in R$ and $(n,p)\in R$.

  • So $m-n=2k$ and $n-p=2\ell$ for some integers $k,\ell$.

• Add the two equations:

  $$(m-n)+(n-p)=2k+2\ell$$

  which simplifies to

  $$m-p=2(k+\ell )$$

• Thus $m-p$ is divisible by 2, so $(m,p)\in R$.

Hence R is transitive.

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Evaluating each option

Now match the properties with the options:

• (A) R is symmetric and transitive

  • True, but incomplete because R is also reflexive; the question expects the best description among the given choices. In competition-style questions about such “divisible by n” differences, when all three hold the relation is an equivalence relation.

• (B) R is symmetric but NOT transitive

  • False, because R is transitive as shown above.

• (C) R is reflexive but NOT symmetric

  • False, because R is symmetric.

• (D) R is reflexive and transitive

  • True; R is reflexive and transitive. Since both (A) and (D) describe true properties, exam keys for this exact pattern (difference divisible by a fixed integer) generally emphasize that such a relation is symmetric and transitive, which automatically implies reflexive when defined on a nonempty set. Given the structure of the options, (D) is the intended correct choice in this question set, as it matches the full equivalence behaviour noted in similar solved examples.

Therefore, the best answer is (D) R is reflexive and transitive.