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 April 16, 2024 by Teachoo

Check whether the relation R in the set Z of integers defined as R = {(𝑎, 𝑏) ∶ 𝑎 + 𝑏 is "divisible by 2"} is reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [0].

Note
: This
is similar
to
Example 5
of NCERT –
Chapter 1 Class 12 Relations and Functions

Question 29 Check whether the relation R in the set Z of integers defined as R = {(𝑎, 𝑏) ∶ 𝑎 + 𝑏 is "divisible by 2"} is reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [0].
R = {(a, b) : 𝑎 + 𝑏 is "divisible by 2"}
Check reflexive
Since a + a = 2a
& 2 divides 2a
Therefore,
2 divides a + a
∴ (a, a) ∈ R,
∴ R is reflexive.
Check symmetric
If 2 divides a + b ,
then 2 divides b + a
Hence, If (a, b) ∈ R, then (b, a) ∈ R
∴ 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 − 2b
a + c = 2(k + p − b)
So, 2 divides (a + c)
∴ If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ 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) : 𝑎 + 𝑏 is "divisible by 2"}
Putting b = 0
R = {(a, 0) : 𝑎 is "divisible by 2"}
So,
[0] = All possible values of a
= {…., −6, −4, −2, 0, 2, 4, 6, ….}

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!