Question 29 - CBSE Class 12 Sample Paper for 2021 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards
Last updated at Oct. 27, 2020 by Teachoo
Check whether the relation R in the set Z of integers defined as R = {(π, π) βΆ π + π is "divisible by 2"} is reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [0].
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Question 29 Check whether the relation R in the set Z of integers defined as R = {(π, π) βΆ π + π is "divisible by 2"} is reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [0].
R = {(a, b) : π + π is "divisible by 2"}
Check reflexive
Since a + a = 2a
& 2 divides 2a
Therefore,
2 divides a + a
β΄ (a, a) β R,
β΄ R is reflexive.
Check symmetric
If 2 divides a + b ,
then 2 divides b + a
Hence, If (a, b) β R, then (b, a) β R
β΄ R is symmetric
Check transitive
If 2 divides (a + b) , & 2 divides (b + c) ,
So, we can write
a + b = 2k
b + c = 2p
Adding (1) & (2)
(a + b) + (b + c) = 2k + 2p
a + c + 2b = 2k + 2p
a + c = 2k + 2p β 2b
a + c = 2(k + p β b)
So, 2 divides (a + c)
β΄ If (a, b) β R and (b, c) β R, then (a, c) β R
Therefore, R is transitive.
Thus, R is an equivalence relation in Z.
Now,
Equivalence class containing 0 i.e. [0]
will be all values of a where one element is 0
Now,
R = {(a, b) : π + π is "divisible by 2"}
Putting b = 0
R = {(a, 0) : π is "divisible by 2"}
So,
[0] = All possible values of a
= {β¦., β6, β4, β2, 0, 2, 4, 6, β¦.}
Made by
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.
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