Question

Let Z be the set of all the integers. Then show that the relation $R=\{\left( a,b \right):a,b\in Z\text{ }and\text{ a-b is divisible by 3}\}$ on Z is an equivalence relation.

Answer 1

Hint: Think of the basic definition of the types of relations given in the question and try to check whether the relation mentioned in the questions satisfies the condition for any type of relation or not.

Complete step by step answer:
Before starting with the solution, let us discuss different types of relations. There are a total of 8 types of relations that we study, out of which the major ones are reflexive, symmetric, transitive, and equivalence relation.
Reflexive relations are those in which each and every element is mapped to itself, i.e., $\left( a,a \right)\in R$ . Symmetric relations are those for which, if \[\left( a,b \right)\in R\text{ }\] then $\left( b,a \right)$ must also belong to R. This can be represented as $aRb\Rightarrow bRa$ . Now, transitive relations are those for which, if $\left( a,b \right)\text{ and }\left( b,c \right)\in R$ then $\left( a,c \right)$ must also belong to R, i.e., $\left( a,b \right)\text{ and }\left( b,c \right)\in R\Rightarrow \left( a,c \right)\in R$ .
Now, if there exists a relation, which is reflexive, symmetric, and transitive at the same time, then the relation is said to be an equivalence relation. For example: let us consider a set A=(1,2). Then the relation {(1,2),(2,1),(1,1),(2,2)} is an equivalence relation.
Now let us start with the solution to the above question. See the integers can be of 3 forms: 3k, 3k+1 and 3k+2.If we want (a-b) to be divisible by 3, we need to select a and b such that they are of the same form out of the 3 forms possible. So, if for (a,b), a and b are of the same form, and for (b,c), b and c are of the same form, then it is obvious that a and c are of the same form too. Hence, the relation is transitive.
Also, if (a-b) is divisible by 3, then (b-a) would also be divisible with quotient being negative in the case of (a-b). Hence, the relation can be said to be symmetrical as well.
Finally, if we select any number from set Z, we will find that the relation $R=\{\left( a,b \right):a,b\in Z\text{ }and\text{ a-b is divisible by 3}\}$ is satisfied, as a-a=0 and 0 is divisible by 3, so it satisfies the condition of a reflexive relation as for all a belonging to the set Z there exists the relation (a,a).
 As we have shown that the relation is symmetric, reflexive, and transitive, we can say that the relation is an equivalence relation.

Note: Remember, a relation can also be called a transitive relation if there exists $aRb$ , but there doesn’t exist any relation $bRc$ . Also, most of the questions as above are either solved by using statements based on observation or taking examples, as we did in the above question.