surjection, injective, bijection.txt

surjection, injective, bijection.txt - 3.50 CHARLIE BARNES...

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3.50 CHARLIE BARNES Theorem (3.50) . Let X be a set, let Sym ( X ) be the set of bijections from X to X , and let represent composition. Then ( Sym ( X ) , ) is a group. Proof. Sym ( X ) = { f : X X | bijections } Well-Definedness Functions are defined by what they do to element in their domains. If f and g are in Sym ( X ), their composition is defined. For x X , g f ( x ) can be determined from f and g , so it is also a well-defined function. Associativity Composition of functions is associative. For any x X and any three functions f,g,h Sym ( X ): h [ g f ]( x ) = h [ g ( f ( x ))] definition of = ( h g )[ f ( x )] = [ h g ] f ( x ) Regardless of which two functions are paired by parenthesis, the definition of explains the order of operations. Closure Lemma (1) . Let f : X X and g : X X be injections. Then g f : X X is an injection. Proof. Let a,b X = D g f such that g f ( a ) = g f ( b ). g f ( a ) = g f ( b ) g ( f ( a )) = g ( f ( b )) definition of f ( a ) = f ( b ) as g is injective a = b as f is injective
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surjection, injective, bijection.txt - 3.50 CHARLIE BARNES...

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