Determine whether the given map \[\phi\] is an isomorphism of the first binary structure with the second. If it is not an isomorphism, why not?
\[<\mathbb{Q},*>\ with\ <\mathbb{Q},*>\ where\ \phi(x)=x^{2}\ for\ x \in \mathbb{Q}\]
I have the one-to-one requirement
\[If\ \phi(x)=\phi(y) then\ x^{2}=y^{2}\]
\[Then\ \sqrt{x^{2}}=\sqrt{y^{2}}\]
So x=y
Thus the map is one-to-one.
Now I need to show that it is onto and I need to show
\[\phi(x*y)=\phi(x)*\phi(y)\]

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i think this is not one to one

for example
\[\phi(2)=\phi(-2)\]

and the mistake is in this line
\[x^2=y^2\iff x=y\] should be
\[x^2=y^2\iff x = \pm y\]

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