(a) Show that the composition of two one-to-one functions, f and g, is one-to-one

(b) Express in terms of .

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- Jan 24th 2010, 04:54 PM450081592Show that f o g is one-to-one
(a) Show that the composition of two one-to-one functions, f and g, is one-to-one

(b) Express in terms of . - Jan 24th 2010, 05:50 PMSudharaka
Dear 450081592,

1) Suppose a,b such that,

and

Then, f[g(a)]=f[g(b)]

g(a)=g(b) since f is an one to one function.

a=b since g is an one to one function.

Therefore

Hence is an one to one function.

2) Supose g(a) = c and f(c) = b,

Therefore,

Since is one to one it is invertible,

-----------A

Since f and g are one to one they are invertible.

and

Therefore -------B

From A and B,

- Jan 24th 2010, 06:29 PMDrexel28
Jeez you two, haha. Try... \circ

Also, for the second one there is a particularly nice result if your functions are both mappings from a set into itself. It follows that (the permutation group on ) from where it follows from basic group theory that .