Question

For the binary operation x₁₀ on set S = {1, 3, 7, 9}, find the inverse of e₃.

Answer 1

A composition table consists of elements which are a result of operation on the set elements.

Here we have the operation, a x₁₀b = remainder of ab divided by 10 where a, b ∈ S. 

For b ∈ S to be an inverse of a ∈ S, a x₁₀b = e, where e is the identity element. 

We know for multiplication operation we have the identity element as 1. 

So e = 1. 

For a = 3, 

3 x₁₀ (inverse of 3) = 1

Remainder of \(\frac{3(i)}{10}\) = 1, i is the inverse of 3.

From the table above, 3 x₁₀ 7 = 1

Hence we can conclude that ‘inverse of 3’ must be 7.