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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, $e=-1$.