CBSE Class 12-science Answered
If A = {1, 2, 3, ..., 9} and R is the relation in A x A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A x A. Prove that R is an equivalence relation. Also, obtain an equivalence class [(2, 5)].
Solution:
Reflexive:
a + b = b + a
i.e. (a, b) R (a, b)
So, R is reflexive.
Symmetric:
Let (a, b) R (c, d)
i.e. a + d = b + c
i.e. c + b = d + a
Therefore, (c, d) R (a, b)
So, R is symmetric.
Transitive:
Let (a, b) R (c, d) & (c, d) R (e, f)
i.e. a + d = b + c & c + f = d + e
Since, a + d = b + c
Therefore, a + d + f = b + c + f
i.e. (a + f) + d = b + (c + f)
i.e. (a + f) + d = b + d + e
i.e. a + f = b + e
So, (a, b) R (e, f).
Thus, R is transitive.
Hence, R is an equivalence relation.
To find an equivalence class [(2, 5)], consider (2, 5) R (a, b)
i.e. 2 + b = 5 + a
i.e. b = 3 + a
As A = {1, 2, 3, ...} and (a, b) in A x A.
[(2, 5)] = {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}