Question

Is addition and multiplication are binary operations on = 1, 2, 3, 4,5 + 5 . If it is
find inverse of each element?

Answer 1

Answer:

Given X={0,1,2,3,4,5}

a∗b={a+b ,if a+b<6a+b−6if a+b≥6}

To check if zero is the identify, we see that a∗0=a+0=a

for all a∈x and also 0∗a=0+a=a for a∈x

Given a∈X,a+0<6 and also 0+a<6

⇒0 is the identity element for the given operation.

Now

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+a ,if a+b<6a+b−6=0=b+a−6if a+b≥6}

i.e, a=−b or b=6−a

but since, ab∈x={0,1,2,3,4,5}a