Question

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 1

It is given that a binary operation ∗ is defined on the set { 0,1,2,3,4,5 } as,

a∗b={ a+b          ,if a+b<6 a+b−6     ,if a+b≥6

Check for a+b<6,

a∗b=b∗a a+b<6

Substitute the value of b=0 in the above inequality, then a<6.

There can be digits less than 6, therefore,

a∗0=a+0=a 0∗a=0+a=a

So, 0is the identity of ∗.

Now, check for a+b≥6.

a∗b=a+b a+b≥6

Substitute the value of b=0 in the above inequality, then a≥6.

This is not possible as value of a can be between ( 0,1,2,3,4,5 ).

The element a∈X is said to be invertible if there exists b∈X ,such that a∗b=0=b∗a.

This means,

a∗b={ a+b=0=b+a               ,if a+b<6 a+b−6=0=b+a−6     ,if a+b≥6

Which implies that, a=−b or b=6−a. Since a≠−b because the set X does not contain negative values, therefore b=6−a is the inverse of a.

Hence, zero is the identity for the given operation and the inverse of the element a is 6−a.