## IB Maths HL Option Sets inverse element

Operation table of a group, Symmetries of plane figures, Cycle notation for permutations, Left and right cosets, group homomorphism, kernel, kernel and range of a homomorphism, isomorphisms

### IB Maths HL Option Sets inverse element

IB Mathematics HL Option: Sets, Relations and groups (inverse element)

Let the binary operation on R defined be x*y=x+y+24. Does each element have an inverse element?

Thanks
elizabeth

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Joined: Mon Jan 28, 2013 8:09 pm

### Re: IB Maths HL Option Sets inverse element

IB Mathematics HL Option Sets, relations and Groups - inverse element

The basic principles about iverse element are:
- An element $x^{-1}$ in a set A is an inverse element for an operation * defined over A if
$x^{-1}*x=x* x^{-1}=e$ for every element $x\in A$
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.
First we are going to find the identity element as follows
$x*e=e*x=x =>x+e+24=x =>e=-24$
suppose that y be a right-inverse element of x,
then
$x*y=x+y+24=e => x+y+24=-24 => x+y=-48 =>y=-x-48$
The right-inverse element y is also a left inverse of x since
$( -x-48)*x=-x-48+x+24=-24=e$
Therefore every element $x$ has an inverse of the form $-x-48$
miranda

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Joined: Mon Jan 28, 2013 8:03 pm