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

Postby elizabeth » Thu Mar 07, 2013 7:45 pm

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?

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Re: IB Maths HL Option Sets inverse element

Postby miranda » Thu Mar 07, 2013 7:55 pm

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
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