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Find if the given operation has identity: $\;\; a \ast b = a^2 + b^2$

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  • An element $e \in N $ is an identify element for operation * if $a \ast e=e \ast a$ for all $a \in N$
Given $ a *b =a^2+b^2$
Let e be an identity element of * operation
By definition of identity element we have a*e=e*a = a
but by definition of * operation $ a *e =a^2+e^2$ =>$ a=a^2+e^2$ =>$ e^2= a-a^2$
=> e = square root of$ a=a^2$, but square root of $a=a^2$ does not always belong to N for all a in N
Therefore the operation * defined by $ a *b =a^2+b^2$ has no identity.
answered Mar 20, 2013 by thagee.vedartham
Operation has no identity
answered Mar 30, 2016 by balaji.thirumalai

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