Question

Let * be defined in Z by: m*n=m+n+2 a) show that every element of Z has an inverse under *. ?

Answer 1

see below

Explanation:

`" * in " ZZ " is defined as: " m"*"n=m+n+2`

(note `+" is commutative in "ZZ`)

to show that each element has an inverse we need to show that

`AA x inZZ " " x**x^(-1) = e`

the first problem is to find `e`

that is by definition

`m**e=m`

`m**e=m+e+2=m`

`=>e=-2`

this is independent of `m` so is the identity for this operation in `ZZ`

to find an inverse

`m in ZZ`

`m**m^(-1)=e=-2`

`m+m^(-1)+2=-2`

`=>m^(-1)=-4-m`

so inverses exist.

eg for `5`

`5^(-1)=-4-5=-9`

test it out

`5**5^(-1)=5+5^(-1)+2=5+ -9+2=-2" "` the identity element.