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