Let A = {1, 2, 3, …, 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also obtain the equivalence class [(2, 5)].

Given that, A = {1, 2, 3, ... 9} and R be the relation in A ×A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A ×A.

Reflexivity:

Let (a,b) R (a,b)

⇒a+b = b+a

which is true since addition is commutative on N.

⇒ R is reflexive.

Symmetricity:

Let (a,b) R (c,d)

⇒a+d = b+c

⇒b+c = a+d

[since addition is commutative on N]

⇒c+b = d+a

⇒ (c,d) R (a,b)

⇒ R is symmetric.

Transitivity:

Let (a,b) R (c,d) and (c,d) R (e,f)

⇒a+d = b+c and c+f = d+e

⇒ (a+d) – (d+e) = (b+c ) – (c+f)

⇒ a-e= b-f

⇒a+f = b+e

⇒ (a,b) R (e,f)

⇒ R is transitive.

Hence, R is an equivalence relation.

Equivalence class [(2,5)]

Let (a,b) R (2,5)

⇒ a+5 = b+2

⇒ a-b = 2-5

= -3

Hence,

The equivalence class [(2,5)] = {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}

In general,

{(a,a+3)|a∈ A}