Question

construct the composition table for the binary operation * defined on {0,1,2,3,4} by a*b=a×b(mod 5),find the identity elements of 2and 4.​

Answer 1

ANSWER

Given X={0,1,2,3,4,5}

a∗b={

a+b ,if a+b<6

a+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<6

a+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



=−b

Hence b=6−a is the inverse of a i.e, a

−1

=6−a

∀a∈{1,2,3,4,5}