NCERT Solutions
Class 12 Maths
Relations and Functions
Misc. Ex. Q14
Define a binary operation *on the set {0, 1, 2, 3, 4, 5} asa ∗ b = 
Show that zero is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.
Let X = {0, 1, 2, 3, 4, 5}.
The operation * on X is defined as a ∗ b = 
An element e ∈ X is the identity element for the operation *, if
a ∗ e = a = e ∗ a for all a ∈ X
For a ∈ X, we have
a ∗ 0 = a + 0 = a [a ∈ X ⇒ a + 0 < 6]
0 ∗ a = 0 + a = a [a ∈ X ⇒ 0 + a < 6]
So, a ∗ 0 = a = 0 ∗ a for all a ∈ X
Thus, 0 is the identity element for the given operation *.
An element a ∈ X is invertible if there exists b ∈ X such that a * b = 0 = b * a. i.e., 
⇒ a = −b or b = 6 − a
But, X = {0, 1, 2, 3, 4, 5} and a, b ∈ X. Then, a ≠ −b
So, b = 6 − a is the inverse of a for all a ∈ X.
Hence, the inverse of an element a ∈X, a ≠ 0 is 6 − a i.e., a-1 = 6 − a.