Q. 94.3( 6 Votes )
Define a new oper
Consider m*n = m2 + n2.
Yes, ℕ 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 ℕ then m2 + n2 also belongs to ℕ.
(ii) Commutative means x*y = y*x
where x and y belongs to ℕ
⇒m*n = m2 + n2
And n*m = n2 + m2
⇒ m2 + n2 = n2 + m2
⇒m*n = n*m
Hence, * is commutative on ℕ
(iii) Associative means (x*y)*z = x*(y*z)
where x, y and z belongs to ℕ
m*n = m2 + n2
(m*n)*o = (m2 + n2)*o = (m2 + n2)2 + o2
Similarly, n*o = n2 + o2
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.
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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