Question

An integer \(\mathrm{m}\) is said to be related to another integer \(\mathrm{n}\) if \(\mathrm{m}\) is a multiple of \(n\) then the relation is (a) Reflexive and symmetric (b) Reflexive and transitive (c) Symmetric and transitive (d) Equivalence relation

Short answer

The given relation is reflexive and transitive, but not symmetric. Therefore, the correct answer is (b) Reflexive and transitive.

Step-by-step solution

Reflexive relation

To verify if the relation is reflexive, we need to check if "every integer is related to itself." In other words, m must be a multiple of m for every integer m. Since any integer m can always be expressed as m * 1, where 1 is an integer, m is always a multiple of itself. So the given relation is reflexive.

Symmetric relation

To verify if the relation is symmetric, we need to check if "for every pair of integers m and n, if m is related to n, then n must be related to m." In other words, if m is a multiple of n, then n must be a multiple of m for every pair of integers m and n. This is not true for every pair of integers, for example, if m=4 and n=2, m is a multiple of n (4 = 2 * 2), but n is not a multiple of m (2 ≠ 4 * k for any integer k). So the given relation is not symmetric.

Transitive relation

To verify if the relation is transitive, we need to check if "for every triplet of integers m, n, and p, if m is related to n and n is related to p, then m must be related to p." In other words, if m is a multiple of n and n is a multiple of p, then m must be a multiple of p for every triplet of integers m, n, and p. Let m = n * x and n = p * y, where x and y are integers. So, m = (p * y) * x = p * (x * y), where x * y is also an integer. Thus, m is a multiple of p. So the given relation is transitive. The relation is Reflexive and Transitive, but not Symmetric. The correct answer is: (b) Reflexive and transitive

Key concepts

Reflexive Relation

In mathematics, a reflexive relation is a relation where every element is related to itself. This means, for any element in a set, when you relate it to itself, the statement should always hold true.

Consider a set of integers. For any integer \( m \), if we say \( m \) is related to \( n \) if \( m \) is a multiple of \( n \), then we need to check if \( m \) is a multiple of \( m \).

This is always true because any number \( m \) can be expressed as \( m \times 1 \), where 1 is also an integer. Therefore, \( m \) is indeed a multiple of itself, making the relation reflexive.

• Example: For \( m = 5 \), since \( 5 = 5 \times 1 \), the reflexive condition is satisfied.

Symmetric Relation

A symmetric relation is defined such that for any pair of elements \( m \) and \( n \), if \( m \) is related to \( n \), then \( n \) should be related to \( m \) as well.

Let's analyze this in our given problem. Suppose \( m \) is a multiple of \( n \), can we confirm if \( n \) is also a multiple of \( m \)? This condition must hold for all integer pairs for the relation to be symmetric.

Unfortunately, this is not true for all cases. For instance, if \( m = 4 \) and \( n = 2 \), 4 is a multiple of 2 (since \( 4 = 2 \times 2 \)). However, 2 is not a multiple of 4 because there is no integer \( k \) such that \( 2 = 4 \times k \). Hence, the given relation is not symmetric.

• Counter Example: \( m = 4 \) and \( n = 2 \) shows asymmetry as 2 is not a multiple of 4.

Transitive Relation

A transitive relation means that if an element \( m \) is related to \( n \), and \( n \) is related to \( p \), then \( m \) must also be related to \( p \).

In terms of multiples, this means if \( m \) is a multiple of \( n \), and \( n \) is a multiple of \( p \), then \( m \) should be a multiple of \( p \).

Let's break it down: Suppose \( m = n \times x \) and \( n = p \times y \), where \( x \) and \( y \) are integers, then \( m = (p \times y) \times x = p \times (x \times y) \). Here, \( x \times y \) is also an integer, proving \( m \) is indeed a multiple of \( p \).

• Example: If \( m = 8 \), \( n = 4 \), and \( p = 2 \), since 8 is a multiple of 4 and 4 is a multiple of 2, then 8 is a multiple of 2, confirming transitivity.