- For each of the sets below and the corresponding binary relation, prove that the relation is binary relation and find the quotient set.
(a) Let A={1,2,3,4,…} be the set of natural numbers. Consider the binary relation R on A defined by: for all n,m∈A, (n,m)∈R if, and only if, their difference n−m is divisible by 10.
So, for this question, I already verified that it is reflexive, symmetric, and transitive. However, I am not sure how to begin to find the quotient set… Won't there be infinity equivalence classes?
Can someone please explain to me how to approach this?
Thanks!
Best Answer
Let's look at the class of $0$ : $$0^\equiv = \{\dots; -20; -10; 0, 10 ; 20 ;\dots\}$$ Now look at the class of $7$ : $$7^\equiv = \{\dots; -13; -3; 7, 17 ; 27 ;\dots\}$$
Each class is infinite, but there will be exactly 10 equivalence classes. They correspond to the different remainders you can get with an Euclidean division by 10.
In other words, $m\equiv n \Longleftrightarrow m \operatorname{Mod}10 = n \operatorname{Mod} 10$.