Get Answers to all your Questions

Name: Please Solve RD Sharma Class 12 Chapter Relation Exercise 1.2 Question 8 Maths Textbook Solution.
Uploaded: Sun, 2021-07-04 18:49
Description: Please Solve RD Sharma Class 12 Chapter Relation Exercise 1.2 Question 8 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter Relation Exercise 1.2 Question 8 Maths Textbook Solution.

Answers (1)

Answer:R is an equivalence relation and the set of all elements related to 1 is 1.
Hint: To prove equivalence relation it is necessary that the given relation should be reflexive, symmetric and transitive.
$Given: \! Set A=\left \{ x \: \epsilon \: Z ;0 \leq x \leq 12 \right \}$
$Also\: given\: that\: relation\: R=\left \{ \left ( a, b \right )\! :\! a=b \right \} is \: defined\: on \: set\: A$
Explanation:
Reflexivity:
Let a be an arbitrary element of A.
Then,
$a=a \; \; \; \; \; \; [Since, every\: element \: is\: equal\: to\: itsel\! f]$
$(a, a) \: \epsilon \: R\: f\! or\: all\: a \: \epsilon \: A$
So, R is reflexive on A
Symmetry:
$Let (a, b)\: \epsilon \: R$
       $a=b$
       $b=a$
      $(b, a) \: \epsilon \: R\: f\! or\: all\: a,b \: \epsilon \: A$
So, R is symmetric on A.
Transitivity:
$Let\: a, b\: and\: (b, c) \: \epsilon \: R$
$a=b ...(i)\: and \: b=c ...(ii)$
multiplying eqn (i) and (ii), we get
      $ab=bc$
      $a=c$
$there\! f\! ore, (a, c) \: \epsilon \: R$
So, R is transitive on A.
Therefore, R is reflexive, symmetric and transitive.
Hence, R is an equivalence relation on A.
$R=\left \{ \left ( a, b \right ):a=b\: \right \} and\: 1 \: is\: on\: element\: o\! f \: A.$
$R=\! \left \{\left ( 1, 1 \right ):1=1 \right \}$
Thus, the set of all elements related to 1 is 1.