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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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