Question 21 (OR 2nd Question) Let R be the relation in the set Z of integers given by R ={(a, b): 2 divides a – b}. Show that the relation R transitive? Write the equivalence class [0].
R = {(a, b) : 2 divides a – b}
Check transitive
If 2 divides (a – b) , & 2 divides (b – c) ,
So, 2 divides (a – b) + (b – c) also
So, 2 divides (a – c)
∴ If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R
Therefore, R is transitive.
Rough
2 divides 8 &
2 divides 12
2 divides 8 + 12 = 20 also
Now,
Equivalance Class [0]
This means that one element is 0, we need to find other elements which satisfy R
R = {(a, b) : 2 divides a – b}
So, (a, 0) ∈ R where a ∈ Z
Because given that R is in set Z, so both a and b belong to set Z
Now,
If (a, 0) ∈ R
2 divides a – 0
i.e. 2 divides a
So, possible values of a are 0, ±2, ± 4, ± 6, …..
i.e. all even numbers and 0
We use plus minus sign because 2 can divide 2 and –2
So, Equivalence Class [0] = {0, ±2, ± 4, ± 6, …..}
Note: 1/2 marks will be deducted if you don’t write
0 or plus minus
in the set

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.