Question

Let R be the relation in the set Z of all integers defined by R={(x,y):x-y is an integer}. Then R is:
(a) Reflexive only
(b) Not transitive
(c) An equivalence relation
(d) Symmetric only

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 if we talk about the relation R={(x,y):x-y is an integer}, we can say that the relation is reflexive. If we subtract a number from itself, the answer is 0 zero is an integer.
Also, if (a-b) is an integer, then (b-a) would also be an integer, which will be negative of the integer that we get in the case of (a-b). Hence, the relation can be said to be symmetrical as well.
Finally, we can say that the relation is transitive too, as the subtraction of two integers always results in an integer. So, no matter whichever numbers we select from the set Z, the relation R={(x,y):x-y is an integer} will always hold true.
As we have shown that the relation is symmetric, reflexive, and transitive, we can say that the relation is an equivalence relation. Hence, the answer to the above question is option (c).

Note: Remember, a relation can also be called a transitive relation if there exists $aRb$ , but there doesn’t exist any relation $bRc$ . For example: a relation $R=\\{(2,3),(4,3),(1,3)\\}$ defined for the set integer is transitive, as it contains a term of the form $aRb$ but no term of the form $bRc$ . 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.