Question

Determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answers. O is the relation defined on Z as follows: For all m, N ∈ Z , mOn⇔m−n is odd.

Answer 1

The given relation O on Z is not reflexive, symmetric, or transitive.

Step-by-step explanation:

The given relation O on Z is defined as follows: For all m, n ∈ Z, mO​n⇔m−n is odd. Let's determine whether the relation is reflexive, symmetric, or transitive.

Reflexive: A relation is reflexive if every element is related to itself. In this case, we need to check if mO​m is true for all m ∈ Z.

Let's consider a case where m = 1. Here, 1O1⇔1−1 is odd, which is not true. Therefore, the relation is not reflexive.

Symmetric: A relation is symmetric if whenever mO​n is true, nOm should also be true. Let's check if nOm is true for all m, n ∈ Z.

Let's consider a case where

m = 2 and

n = 3.

Here, 2O3⇔2−3 is odd, which is true.

However, 3O2⇔3−2 is not odd, which is false. Therefore, the relation is not symmetric.

Transitive: A relation is transitive if whenever mO​n is true and nOp is true, then mOp should also be true. Let's check if mOp is true for all m, n, p ∈ Z.

Let's consider a case where

m = 1,

n = 2, and

p = 3.

Here, 1O2⇔1−2 is odd, which is true, and 2O3⇔2−3 is odd, which is true.

However, 1O3⇔1−3 is not odd, which is false. Therefore, the relation is not transitive.

Based on our analysis, the given relation O on Z is none of these (reflexive, symmetric, transitive).