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Home » Show that the relation R defined on the set A = (1, 2, 3, 4, 5), given by R = {(a, b) : |a – b| is even} is an equivalence relation.
Show that the relation R defined on the set A = (1, 2, 3, 4, 5), given by
R = {(a, b) : |a – b| is even} is an equivalence relation.
CBSE | Class 12 | Exercise 1B | Maths | Relations | RS Aggarwal
Show that the relation R defined on the set A = (1, 2, 3, 4, 5), given by
R = {(a, b) : |a – b| is even} is an equivalence relation.

Solution:

\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathrm{A} and |\mathrm{a}-\mathrm{b}| is even \} where \mathrm{A}=
(As given)
If \mathrm{R} is Reflexive, Symmetric and Transitive, therefore \mathrm{R} equivalence relation.

Reflexivity:
Suppose a be an arbitrary element of A
|\mathrm{a}-\mathrm{a}|=0
As 0 is even
\Rightarrow(\mathrm{a}, \mathrm{a}) \in \mathrm{R}
Therefore, \mathrm{R} is reflexive.

Symmetric:
Suppose a and b \in A, such that (a, b) \in R
\begin{array}{l} \Rightarrow|\mathrm{a}-\mathrm{b}|=\text { even } \\ \Rightarrow|\mathrm{b}-\mathrm{a}|=\mathrm{even} \\ \Rightarrow(\mathrm{b}, \mathrm{a}) \in \mathrm{R} \end{array}
Therefore, R is symmetric.

Transitivity:
Suppose a, b and c \in A, such that (a, b) \in R and (b, c) \in R
\Rightarrow|\mathrm{a}-\mathrm{b}| is even and |\mathrm{b}-\mathrm{c}| is even
This is only possible when a and b both are even or odd and \mathrm{b} and \mathrm{c} both are even or odd.
Two cases arise:

Case I: If b is an even
Suppose (a, b) \in R and (b, c) \in R
\Rightarrow|a-b| is even and |b-c| is even
As b is even
\Rightarrow a is even and c is even
\Rightarrow|\mathrm{a}-\mathrm{c}| is even
\Rightarrow(a, c) \in R

Case II: If b is an odd
Suppose (\mathrm{a}, \mathrm{b}) \in \mathrm{R} and (\mathrm{b}, \mathrm{c}) \in \mathrm{R}
\Rightarrow|\mathrm{a}-\mathrm{b}| is even and |\mathrm{b}-\mathrm{c}| is even
\Rightarrow As b is odd
\Rightarrow \mathrm{a} is odd and \mathrm{c} is odd
As difference of two odd numbers is even
\Rightarrow|\mathrm{a}-\mathrm{c}| is even
\Rightarrow(\mathrm{a}, \mathrm{c}) \in \mathrm{R}
Therefore, R is transitive.
As a result, R is an equivalence relation.

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