Question

Let $R=\left\{ \left( a,b \right):a,b\in Z\text{ }and\text{ }\left( a-b \right)\text{ }is\text{ }divisible\text{ }by\text{ }5 \right\}$. Show that R is an equivalence relation on Z.

Answer 1

Hint: In this question, we are asked to prove a relation R as an equivalent relation, for which we will prove the relation as reflexive, symmetric and transitive because equivalent relation is one that satisfies the conditions of reflexive, symmetric and transitive relation. Also, we need to remember that 0 is divisible by all the numbers.

Complete step-by-step answer:
In this question, we have been asked to prove that relation $R=\left\{ \left( a,b \right):a,b\in Z\text{ }and\text{ }\left( a-b \right)\text{ }is\text{ }divisible\text{ }by\text{ }5 \right\}$ is an equivalence relation. So, to prove this, we will prove that the given relation R is reflexive, symmetric, and transitive relation because we know that any relation is equivalent only when it is reflexive, symmetric, and transitive relation.
Now, let us first go with reflexive relation. Reflexive relation is a relation which maps for itself. So, for, $R=\left\{ \left( a,a \right):a,b\in Z \right\}$, we have to prove that a - a is divisible by 5. We know that a - a will give the answer as 0 and we know that 0 is divisible by all the numbers, hence 0 is divisible by 5.
Therefore, we can say that relation R is a reflexive relation.
Now, let us go with the symmetric relation. Symmetric relation states that if $aRb$ then $bRa$. Here we have been given relation $R=\left\{ \left( a,b \right):a,b\in Z\text{ }and\text{ }\left( a-b \right)\text{ }is\text{ }divisible\text{ }by\text{ }5 \right\}$. So, if it is true, then we have to prove that R is true for $\left\{ \left( b,a \right):a,b\in Z \right\}$. Now, we have been given that a - b is divisible by 5, which means that we can write a - b = 5m. So, we can write it as b - a = - 5m. And, we can write (- 5m) as 5(- m), which is again divisible by 5. Hence, we can say that (b - a) is divisible by 5, that is R is true for $\left\{ \left( b,a \right):a,b\in Z \right\}$.
Therefore, we can say that relation R is a symmetric relation.
Now, let us go with a transitive relation, which states that $aRb$ and $bRc$, then $aRc$. So, to check if the relation is transitive or not, we will consider a, b and c. So, let us consider $R=\left\{ \left( a,b \right):a,b\in Z\text{ }and\text{ }\left( a-b \right)\text{ }is\text{ }divisible\text{ }by\text{ }5 \right\}$ and $R=\left\{ \left( b,c \right):b,c\in Z\text{ }and\text{ }\left( b-c \right)\text{ }is\text{ }divisible\text{ }by\text{ }5 \right\}$ satisfies, now we will check whether $aRc$ is satisfied or not. We know that (a - b) is divisible by 5 and we can write it as,
a - b = 5m ………… (i)
We also know that (b - c) is divisible by 5, so we can write it as,
b - c = 5n ………… (ii)
Now, we will add equation (i) and (ii). So, we get,
a - b + b - c = 5m + 5n, which is a - c = 5 (m + n). And if we consider m + n = x, then we can write a - c = 5x, which means that (a - c) is divisible by 5.
Hence, we can say that R is a transitive relation.
Therefore, we have proved that relation R is reflexive, symmetric, and transitive relation. Hence, relation R is an equivalent relation.

Note: While solving this question, some of us might get confused with - 5m, that is, whether it is divisible by 5 or not, or whether this should be included or not, but you have to remember that we are talking about integers here, which may give negative answer and so - 5m is divisible by 5.