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Name: Please Solve RD Sharma Class 12 Chapter Relation Exercise 1.2 Question 12 Maths Textbook Solution.
Uploaded: Sun, 2021-07-04 22:24
Description: Please Solve RD Sharma Class 12 Chapter Relation Exercise 1.2 Question 12 Maths Textbook Solution.

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

Answers (1)

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}