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.


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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