Question

Define a relation on \(\mathbb{Z}\) by declaring \(x R y\) if and only if \(x\) and \(y\) have the same parity. Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why. What familiar relation is this?

Short answer

The relation \(R\) is reflexive, symmetric, and transitive. This is the equivalence relation called equality modulo 2.

Step-by-step solution

Check Reflexivity

Every integer \(x \in \mathbb{Z}\) has the same parity as itself, so \(xRx\) holds true for all \(x \in \mathbb{Z}\). Therefore, the relation \(R\) is reflexive.

Check Symmetry

If \(xRy\), then \(x\) and \(y\) have the same parity. But this means that \(y\) and \(x\) also have the same parity, so \(yRx\) holds true. Therefore, the relation \(R\) is symmetric.

Check Transitivity

If \(xRy\) and \(yRz\), then \(x\), \(y\), and \(z\) all have the same parity. So \(xRz\) also holds true. Therefore, the relation \(R\) is transitive.

Identify the Relation

The relation \(R\) is the equivalence relation called equality modulo 2, which partitions every integer into one of two classes: one class containing the even numbers, and the other class containing the odd numbers.