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

Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \(a * b =\begin{cases}a + b, & if \; a + b < 6\\a + b - 6 &if \; a + b \geq 6\end{cases}\)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.

From the composition table it is obvious that

a * 0 = a ∀ a ∈ {0, 1/2, 3,4, 5}

∴ 0 is the identity element

Also ∀ a ∈ {0, 1, 2, 3, 4, 5} ∃ (6 – a) ∈ {0,1,2, 3,4,5} such that

a * (6 – a) = (6 – a) * a = 0, a ≠ 0

hence 6 – a is the inverse of a .

However when a = 0, 6 – a g {0, 1, 2, 3,4, 5}

hence 6 – a is the inverse of a when a ≠ 0.