Question

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

a * b = `{(a+b, "if a+b < 6"), (a + b - 6, if a +b >= 6):}`

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:

a * b = `{ (a+b, "if a+b < 0"), (a+ b - 6, "if a + b >= 6"):}`

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

For a ∈ X we observed that

a * 0 = a + 0 =a   [a ∈ X ⇒ a + 0 < 6]

0 * a = 0 + a = a        [a ∈ X ⇒ 0 + a < 6]

:. a * 0 =a = 0 * a ∀ 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.

ie `{(a+b = 0= b + a, if a + b < 6),(a+6 - 6= 0=b + a - 6, if a+b >= 6):}`

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.