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
$a*b=\left\{\begin{array}{ll}a+b&if\:a+b<6\\a+b-6&if\:a+b\geq6\end{array}\right.$

An element e ∈ X is the identity element for the operation *,
If a * e = a = e * a ∀ a ∈ X
For a ∈  X, we can see that:
a * 0 = a + 0 = a [a ∈  X = > a + 0 < 6]
0 * a = 0 + a = a [a ∈  X = > a + 0 < 6]
⇒ a * 0 = a = 0 * a ∀ a ∈  X.
Therefore, 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.
$\implies\left\{\begin{array}{ll}a+b = 0 = b + a&if\:a+b<6\\a+b-6 = 0 = b + a - 6&if\:a+b\geq6\end{array}\right.$
so, a = -b and b = 6 - a
But, X = {0, 1, 2, 3, 4, 5} and a, b ∈ X. Then, a ≠ -b.
Therefore, b = 6 – a is the inverse of a ∈ X.
Thus, the inverse of an element a ∈ X, a ≠ 0 is 6 – a, a⁻¹ = 6 – a.