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Problem 2
The map $\phi: \mathbb{Z} \rightarrow \mathbb{Z}$ defined by $\phi(n) = n+1$ is one-to-one and onto $\mathbb{Z}$. Give the definition of a binary operation $*$ on $\mathbb{Z}$ such that $\phi$ is an isomorphism mapping
(a) $\langle\mathbb{Z}, +\rangle$ onto $\langle\mathbb{Z}, *\rangle$.
(b) $\langle\mathbb{Z}, *\rangle$ onto $\langle\mathbb{Z}, +\rangle$.
In each case, give the identity element of $\langle\mathbb{Z}, *\rangle$.
Solution
(a) It is already stated that $\phi$ defines a function that is one to one and onto, meaning that all we have to show is the homomorphism property on $\langle\mathbb{Z}, +\rangle$ and $\langle\mathbb{Z}, *\rangle$. To satisfy the homomorphism property, where $x,y \in \mathbb{Z}$,
(1)Now, by letting $*$ be define by $x*y = x+y-1$.
(3)Now, finding the identity
(5)Hence, $e=1$.
(b) It is already stated that $\phi$ defines a function that is one to one and onto, meaning that all we have to show is the homomorphism property on $\langle\mathbb{Z}, *\rangle$ and $\langle\mathbb{Z}, +\rangle$. To satisfy the homomorphism property, where $x,y \in \mathbb{Z}$,
(7)Now, by letting $*$ be define by $x*y = x+y+1$.
(9)Now, finding the identity
(11)Hence, $e=-1$.