Question

Show that the relation R in the set A = (1,2,3,4,5) given by $R=\\{\left( a,b \right):|a-b|\text{ is even }\\!\\!\\}\\!\\!\text{ }$ , is an equivalence relation. Show that all elements of {1,3,5} are related to each other and all elements of {2,4} are related to each other. But no element of {1,3,5} is related to any element of {2,4}.

Answer 1

Hint: Any relation can be classified as reflexive, symmetric and transitive. If aRa exists in the relation, then it is said to be reflexive. If aRb and bRa both exist in the relation, then it is said to be symmetric. If aRb and bRc exist implies that aRc also exists, the relation is transitive. Also, use the property that modulus of the difference between two whole numbers is even if both the numbers are either even else odd. Using these definitions we will check the relation R.

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. The given set A=(1,2,3,4,5), and the relation is $R=\\{\left( a,b \right):|a-b|\text{ is even }\\!\\!\\}\\!\\!\text{ }$ . See if we select any number from set A, we will find that the relation is true as |a-a|=0 and 0 is even, so it satisfies the condition of a reflexive relation as for all a belonging to the set A there exists the relation (a,a).

Now, if we observe we will find that the relation is symmetric as well. As for being a symmetric relation, if $|a-b|$ is even then $|b-a|$ must also be even, and this will be always true in case of set A, as (b-a)=-(a-b) and when we take modulus on both sides, we find that $|a-b|=|b-a|$ .

Further, the relation is transitive too, as for $|a-b|$ to be even, a and b both must be either even else both must be odd, so let us take a and b to be odd. Therefore, $|b-c|$ to be even, provided b is odd, c must be odd too. So, as a, b and c are odd $|a-c|$ is even satisfying the condition. Hence, the relation is transitive as well.

Hence, we have shown that the relation $R=\\{\left( a,b \right):|a-b|\text{ is even }\\!\\!\\}\\!\\!\text{ }$ for set A=(1,2,3,4,5), is reflexive, symmetric and transitive. So, we can say that the relation is an equivalence relation.

Also, modulus of the difference of two whole numbers is even if both the numbers are either even or both numbers odd. So, all elements of {1,3,5} are related to each other and all elements of {2,4} are related to each other. But no element of {1,3,5} is related to any element of {2,4}.

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. Also if it is not possible to prove that relation is symmetric, reflexive or transitive, then use a suitable example to show that it is not.