A binary operation * on the set 0, 1, 2, 3, 4, 5 is defined as: a… - Vidyadip

Question

A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as: a * b = $ \begin{matrix} \text{a + b} & \text{if} & \text{a + b < 6} \\ \text{a + b - 6,} & \text{if} & \text{a + b }\geq6 \\ \end{matrix}$.
Show that zero is the identity for this operation and each element 'a' of the set is, invertible with 6 – a, being the inverse of 'a'.

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Answer

since a * 0 = a + 0 = a and 0 * a = 0 + a = aNote: $\forall$ a $\in$ {0, 1, 2, 3, 4, 5}
$\therefore$ 0 is the identity for *.
Also,$\forall$ a $\in$ {0, 1, 2, 3, 4, 5}, a * (6 – a) = a + (6 – a) – 6= 0 (which is identity)
Each element ‘a’ of the set is invertible with (6 – a), being the inverse of ‘a’.

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