Question 1 Let A ={1, 2, 3, 4}. Let R be the equivalence relation on A × A defined by (a, b)R(c, d) iff a + d = b + c . Find the equivalence class [(1, 3)]. Given A = {1, 2, 3, 4}
R is defined as
(a, b)R(c, d) iff a + d = b + c
In this relation
(a, b) goes in , and (c, d) comes out.
We need to find [(1, 3)]
So, (1, 3) will go in, and (c, d) will come out
This will be possible if
a + d = b + c
1 + d = 3 + c
d – c = 3 – 1
d – c = 2
So, in our relation [(1, 3)]
We need to find values of c and d which satisfy d – c = 2
Since (c, d) ∈ A × A
Both c and d are in set A = {1, 2, 3, 4}
d – c
Numbers
(c, d)
2 – 1 = 1
d = 2, c = 1
Not possible
3 – 1 = 2
d = 3, c = 1
(1, 3)
3 – 2 = 1
d = 3, c = 2
Not possible
4 – 1 = 3
d = 4, c = 1
Not possible
4 – 2 = 2
d = 4, c = 2
(2, 4)
4 – 3 = 1
d = 4, c = 3
Not possible
So, only (1, 3) and (2, 4) satisfy
∴ [(1, 3)] = { (1, 3), (2, 4) }

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.