Answer :

Consider  m*n = m2 + n2.
(i)

Yes,
is closed under *.
This is because, for any natural number m,
m2 is also a natural number.
Further, on adding two natural numbers, we get a natural number only.
So, if m and n belong to
then m2 + n2 also belongs to .


(ii) Commutative means x*y = y*x
where x and y belongs to

m*n = m2 + n2

And n*m = n2 + m2

m2 + n2 = n2 + m2

m*n = n*m

Hence, * is commutative on


(iii) Associative means (x*y)*z = x*(y*z)
where x, y and z belongs to

m*n = m2 + n2

Further,

(m*n)*o = (m2 + n2)*o = (m2 + n2)2 + o2

Similarly, n*o = n2 + o2

Further,

m*(n*o) = m*(n2 + o2) = m2 + (n2+ o2)2

(m2 + n2)2 + o2 ≠ m2 + (n2 + o2)2

(m*n)*o ≠ m*(n*o)

Hence, * is not associative.

(iv) An identity element is a special type of element of a set with respect to a binary operation on that set,
which leaves other elements unchanged when combined with them.

that is for any element a if a * b = b * a = a
then b is the identity of a.

For *, let k be the identity element
then m*k = m2 + k2 =  k2 + m2  =  m2

This is possible only when k = 0 but k needs to be a natural number.

Hence, * does not have an identity element.

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