Q. 14
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 :
Let X = {0, 1, 2, 3, 4, 5}
The operation * on X is defined as
An element e ϵ X is the identity element for the operation *,
If a * e = a = e * a 
a ϵ X.
For a ϵ X, we can see that:
a * 0 = a + 0 = a [a ϵ X = > a + 0 < 6]
0 * A = 0 + a = a [a ϵ X = > a + 0 < 6]
⇒ a * 0 = a = 0 * a 
a ϵ X.
Therefore, o is the identity element for the given operation *.
An element a ϵ X is invertible if there exists b ϵ X such that
a * b = 0 = b * a.
a = -b or b = 6 – a
But, X = {0, 1, 2, 3, 4, 5} and a, b ϵ X. Then, a ≠ -b.
Therefore, b = 6 – a is the inverse of a ϵ X.
Thus, the inverse of an element a ϵ X, a ≠ 0 is 6 – a, a-1 = 6 – a.
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