Question

Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as

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.

Answer 1

Let X = {0, 1, 2, 3, 4, 5}.

The operation * on X is defined as:

An element e ∈ X is the identity element for the operation *, if

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.

∴b = 6 − a is the inverse of a &mnForE; a ∈ X.

Hence, the inverse of an element a ∈X, a ≠ 0 is 6 − a i.e., a⁻¹ = 6 − a.