#### Please Solve RD Sharma Class 12 Chapter Relation Exercise 1.2 Question 12 Maths Textbook Solution.

Answer: R is an equivalence relation on A.
No number of the subset {1, 3, 5, 7} is related to any number of the subset {2, 4, 6}.
Hint: To prove an equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.
$Given: \! \left \{ A=1, 2, 3, 4, 5, 6, 7 \right \}$
$R=\left \{\left ( a, b \right ) : both\: a\: and \: b\: are\: either \: odd \: or\: even\: number \right \}$
Explanation:
$R=\left \{ \left ( 1,1 \right ),\left ( 1 ,3\right ),\left (1, 5 \right ) , (1,7), \left ( 3, 1\right ),\left ( 3, 3 \right ), \left ( 3, 5 \right ),\left ( 3, 7 \right ), \left (5, 1 \right ),\left ( 5, 3 \right ),\left (5, 5 \right ),\left ( 5, 7 \right ),\left ( 7, 1 \right ) ,\left ( 7, 3 \right ),\left ( 7, 5 \right )\left (7,7 \right ), \left ( 2, 2\right ), \left (2, 4 \right ),\left ( 2, 6 \right ),\left (4, 2 \right ),\left ( 4, 4 \right ) ,\left (4, 6 \right ),(6, 2)\left ( 6, 4 \right ),\left ( 6, 6 \right ) \right \}$We observe the following properties of R on A.
Reflexivity:
$Clearly \: \left (1,1 \right ) ,\left ( 2,2 \right ), \left ( 3, 3 \right ),\left ( 4, 4 \right ),\left (5, 5 \right ) ,\left ( 6,6 \right ),\left ( 7,7 \right ) \: \epsilon \: R$.
So, R is a reflexive relation in A
Symmetric:
$Let \: a, b \: \epsilon \: A\: \text{ be such that (a,b) }\: \epsilon \: R.$
$Then \: (a, b)\: \epsilon \: R$
Both a and b are either odd or even.
Both b and a are either odd or even.
$(b, a) \epsilon \: R$
$Thus, (a, b) \epsilon \: R$
$\Rightarrow (b, a) \epsilon \: R\: f\! or\: al\! l\: a, b \: \epsilon \: A$
So, R is a symmetric relation on A
Transitivity:
$Let \: a, b, c \: \epsilon \: A\: be\: such \: that (a, b) \: \epsilon \: R, (b, c) \: \epsilon \: R$
$Then (a, b) \epsilon \: R$
Both a and b are either odd or even.
$(b, c) \epsilon \: R$
⇒  Both b and c are odd or even.
$\\\text{If both a and b are even, then (b, c) }\epsilon \: R\: \text{ both b and c are even}$$\\\text{If both a and b are odd, then (b, c) }\epsilon \: R\: \text{ both b and c are odd}$
⇒ Both a and c are even or odd.
$\therefore (a, c) \epsilon \: R$
$So, (a, b) \epsilon \: R \: \text{ and (b, c) }\epsilon \: R$
$(a, c) \epsilon \: R$
R is a transitive relation on A
Thus, R is reflexive, symmetric and transitive.
Hence, R is an equivalence relation on A.
We observe that two numbers in A are related if both are odd or both are even.
Since, {1, 3, 5, 7} has all odd numbers of A
So, all the numbers of {1, 3, 5, 7} are related to each other
Similarly, all the numbers of {2, 4, 6} are related to each other as it contains all even numbers of set A.
An even, odd number in A is related to an even, odd number in A respectively.
So, no number of the subset {1, 3, 5, 7} is related to any number of the subset {2, 4, 6}