Define the relation \(\sim\) on \(\Z\) by \(a \sim b\) if and only if \(5 \mid b-a\text{.}\) Prove that \(\sim\) is an equivalence relation and describe the equivalence classes.
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Handout Monday 10/20
Last time we explored the concepts of partitions and equivalence relations. Let’s look at an example of this.
Example 1.
The next learning target asks you to prove that a relation is an equivalence relation and give the equivalence classes. There is some practice in the textbook for this.
Notice in the example we have given, we get five sets as the partition. One of these sets is a subgroup of \(\Z\) (which one). The others are not. But the others can be described in terms of this subgroup. This is what we are really after here.
In fact, could we describe the structure of these five sets using tools we have already developed? Might these five sets interact with each other in a group or ring sort of way? What might this even mean?
