Question

Consider the partition \(P=\\{\\{\ldots,-4,-2,0,2,4, \ldots\\},\\{\ldots,-5,-3,-1,1,3,5, \ldots\\}\\}\) of \(\mathbb{Z}\) Let \(R\) be the equivalence relation whose equivalence classes are the two elements of \(P\). What familiar equivalence relation is \(R\) ?

Short answer

The familiar equivalence relation \(R\) is the parity relationship.

Step-by-step solution

Understand the Partition \(P\)

The partition \(P\) of \(\mathbb{Z}\) is defined as two sets: the set of all even integers \(\{\ldots,-4,-2,0,2,4, \ldots\}\) and the set of all odd integers \(\{\ldots,-5,-3,-1,1,3,5, \ldots\}\). It separates the integers into two groups.

Understand Equivalence Relation

Equivalence relations group elements together based on a certain relationship. In this case, the equivalence relation \(R\) groups the integers based on whether they are even or odd. So two elements would be in relation \(R\) if they belong to the same group, i.e., they are both even or they are both odd.

Identify the Equivalence Relation

The equivalence relation \(R\) defined by the partition \(P\) corresponds to the familiar relation of 'parity'. Parity is the property of an integer's being odd or even.