IB Mathematics HL Option Sets, relations and Groups - inverse element
The basic principles about iverse element are:
- An element
in a set A is an inverse element for an operation * defined over A if
for every element
where e is the identity element.
- For an associative operation *, an element x admits a left-inverse x’and a right-inverse x’’, then these two identities are equal.
- For an operation * on a set A having an identity element then every invertible element admits a unique inverse.
Concerning your question,
First we are going to find the identity element as follows
suppose that y be a right-inverse element of x,
The right-inverse element y is also a left inverse of x since
Therefore every element
has an inverse of the form