Difference between revisions of "Inverse"
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* [[Logical inverse]] | * [[Logical inverse]] | ||
| − | * Inverse with respect to an [[operation]], such as in a [[group]] (see also [[identity]]) | + | * Inverse with respect to an [[operation]], such as in a [[group]] (see also [[identity]]) (see below) |
{{disambig}} | {{disambig}} | ||
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| + | What should the page which the third item above links to be called? Here is some content for it, but I don't know what to call the page: | ||
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| + | Suppose we have a [[binary operation]] G on a set S, <math>G:S\times S \to S</math>, and suppose this operation has an [[identity]] e, so that for every <math>g\in S</math> we have <math>G(e, g) = G(g, e) = g</math>. An '''inverse to g''' under this operation is an element <math>h \in S</math> such that <math>G(h, g) = G(g, h) = e</math>. | ||
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| + | If our operation is not [[commutative]], we can talk separately about ''left inverses'' and ''right inverses''. A left inverse of g would be some h such that <math>G(h, g) = e</math> while a right inverse would be some h such that <math>G(g, h) = e</math>. | ||
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| + | ==Uniqueness (under appropriate conditions)== | ||
| + | If the operation G is [[associative]] and an element has both a right and left inverse, these two inverses are equal. | ||
| + | ===Proof=== | ||
| + | Let g be the element with left inverse h and right inverse h', so <math>G(h, g) = G(g, h') = e</math>. Then <math>G(G(h, g), h') = G(e, h') = h'</math>, by the properties of e. But by associativity, <math>\displaystyle G(G(h, g), h') = G(h, G(g, h')) = G(h, e) = h</math>, so we do indeed have <math>h = h'</math>. | ||
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| + | ===Corollary=== | ||
| + | If the operation G is associative, inverses are unique. | ||
Revision as of 09:33, 13 July 2006
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What should the page which the third item above links to be called? Here is some content for it, but I don't know what to call the page:
Suppose we have a binary operation G on a set S, 
, and suppose this operation has an identity e, so that for every 
we have 
. An inverse to g under this operation is an element 
such that 
.
If our operation is not commutative, we can talk separately about left inverses and right inverses. A left inverse of g would be some h such that 
while a right inverse would be some h such that 
.
Uniqueness (under appropriate conditions)
If the operation G is associative and an element has both a right and left inverse, these two inverses are equal.
Proof
Let g be the element with left inverse h and right inverse h', so 
. Then 
, by the properties of e. But by associativity, 
, so we do indeed have 
.
Corollary
If the operation G is associative, inverses are unique.
