Question

Let S be the set of all real numbers. A relation R has been defined on S by $aRb \Leftrightarrow \left| {a - b} \right| \leqslant 1$ , then R is which of the following?
A) Reflexive and transitive but not symmetric.
B) An equivalence relation.
C) Symmetric and transitive but not reflective.
D) Reflexive

Answer 1

Hint: A relation R is said to be reflexive if every element of A should be related to itself i.e. $\left( {a,a} \right) \in R\forall a \in A$. A relation R is said to be symmetric if \[\left( {a,b} \right) \in R \Rightarrow \left( {b,a} \right) \in R\]
A relation R is said to be transitive if \[\left( {a,b} \right) \in R,\left( {b,c} \right) \in R \Rightarrow \left( {a,c} \right) \in R\].

Complete step by step answer:
Given to us a relation R defined on S such that $aRb \Leftrightarrow \left| {a - b} \right| \leqslant 1$ defined on a set of real numbers.
For a relation R to be reflexive every element of A should be related to itself i.e. $\left( {a,a} \right) \in R\forall a \in A$.
Now, we can write the given relation as $ - 1 \leqslant a - b \leqslant 1$.
In given inequality if $a \in R$ then $b \in R$ and this satisfies the above reflexive relation property i.e. every element that satisfies a also satisfies b.
Therefore our given relation is reflexive.

A relation R is said to be symmetric if \[\left( {a,b} \right) \in R \Rightarrow \left( {b,a} \right) \in R\] , that is \[aRb \Rightarrow bRa\;\forall \left( {a,b} \right) \in R\]
From the given relation, $ - 1 \leqslant a - b \leqslant 1$ which also gives $ - 1 + b \leqslant a \leqslant 1 + b$ or $ - 1 + a \leqslant b \leqslant 1 + a$
From this inequality, it is clear to us that for every value of a there exist multiple values of b and vice versa. So the given relation is not symmetric.

A relation R is said to be transitive if \[\left( {a,b} \right) \in R,\left( {b,c} \right) \in R \Rightarrow \left( {a,c} \right) \in R\]
In the given inequality $ - 1 \leqslant a - b \leqslant 1$ for every value of a there exists multiple values of b and these values become c.
Hence, for every \[\left( {a,b} \right) \in R,\left( {b,c} \right) \in R \Rightarrow \left( {a,c} \right) \in R\]
Therefore, the relation R is transitive.
Hence the given relation R is reflexive and transitive but not symmetric i.e. option (A).

Note: Full forms of the notations used in the above solution:
$ \Leftrightarrow $ - If and only if, $ \in $ - Belongs to, $\forall $ - for all. It is to be noted that these notations are important and to be used while representing a relation between elements/sets.