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 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.

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