Question

Q. A binary operation * on the set [0,1,2,3,4,5] is defined as
a * b = {a+b, if a+b<6a+b-6, if a+b≥6 {a+b, if a+b<6a+b-6, if a+b≥6
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'.

Answer 1

:

Given the set X={0,1,2,3,4,5} where the binary operation ∗ is defined by a∗b={a+ba+b−6ifa+b<6ifa+b≥6

An element e∈N is an identify element for operation * if a∗e=e∗a for all a∈N

To check if zero is the identity, we see that a∗0=a+0=afora∈x and also 0∗a=0+a=afora∈x

Given a∈X,a+0<6 and also 0+a<6

⇒0 is the identify element for the given given operation

Step 2:

The element a∈X is invertible if there exist b∈X such that a∗b=e=b∗a

In this case, e=0→a∗b=0=b∗a.

⇒a∗b={a+b=0=b+aa+b−6=0=b+a−6ifa+b<6a+b≥6

ie a=−borb=6−a

but since a,b∈X={0,1,2,3,4,5}, a≠−b

Hence b=6−a is the inverse of a, i.e., a−1=6−a,∀a∈{1,2,3,4,5