Hw3 Problem 2

### 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)
\begin{align} \phi(x+y) = \phi(x) * \phi(y) \end{align}
(2)
$$(x+y)+1 = (x+1)*(y+1).$$

Now, by letting $*$ be define by $x*y = x+y-1$.

(3)
$$(x+y)+1 = [(x+1)+(y+1)]-1$$
(4)
$$(x+y)+1 = (x+y)+1$$

Now, finding the identity

(5)
$$e*x = x*e = x$$
(6)
$$1*x = x*1= (1+x)-1 = (x+1)-1= x$$

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)
\begin{align} \phi(x*y) = \phi(x) + \phi(y) \end{align}
(8)
$$(x*y)+1 = (x+1)+(y+1).$$

Now, by letting $*$ be define by $x*y = x+y+1$.

(9)
$$(x+y+1)+1 = [(x+1)+(y+1)]$$
(10)
$$(x+y)+2 = (x+y)+2$$

Now, finding the identity

(11)
$$e*x = x*e = x$$
(12)
$$-1*x = x*-1= ((-1)+x)+1 = (x+(-1))+1= x$$

Hence, $e=-1$.