Question

If R is a relation defined as aRb, iff \[\left| {a - b} \right| > 0\] , then the relation is
A. Reflexive
B. Symmetric
C. Transitive
D. Symmetric and Transitive

Answer 1

Hint: A relation R on a set A is said to be an equivalence relation if and only if the relation R is reflexive, symmetric and transitive.
Reflexive: A relation is said to be reflexive, if \[(a,a) \in R\] for every \[a \in R\] .
Symmetric: A relation is said to be symmetric, if \[(a,b) \in R\] then \[(b,a) \in R\] .
Transitive: A relation is said to be transitive if \[(a,b) \in R\] and \[(b,c) \in R\] then \[(a,c) \in R\] .

Complete step by step answer:
Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other, if and only if they belong to the same equivalence class.
We are given that R is a relation defined as aRb, iff \[\left| {a - b} \right| > 0\] . we will check the following properties :
Reflexibility: Let \[a\] be any arbitrary element then \[\left| {a - a} \right| = 0\]
This implies \[a\] is not related to \[a\].
Therefore the relation R is not a reflexive relation.
Symmetry: Let \[a\] and \[b\] be two distinct elements , then \[(a,b) \in R\] means that \[\left| {a - b} \right| > 0\]
Which implies \[\left| {b - a} \right| > 0\]
(because \[\left| {a - b} \right| = \left| {b - a} \right|\] )
Therefore \[(b,a) \in R\]
Thus \[(a,b) \in R\] implies \[(b,a) \in R\]
Therefore the relation R is a symmetric relation .
Transitivity: Let \[(a,b) \in R\] and \[(b,c) \in R\]
Therefore we have \[\left| {a - b} \right| > 0\] and \[\left| {b - c} \right| > 0\]
Now by adding both the equations we get \[\left| {a - b + b - c} \right| > 0\]
Which implies \[\left| {a - c} \right| > 0\]
Therefore \[(a,c) \in R\]
Therefore the relation R is a transitive relation.
Hence we conclude that relation R is a Symmetric and Transitive relation but it is not a Reflexive relation.

So, the correct answer is “Option D”.

Note: : A relation R on a set A is said to be an equivalence relation if and only if the relation R is reflexive, symmetric and transitive. If any of these properties does not hold true then the relation R is never an equivalence relation . Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes.