(a) Show that the composition of two one-to-one functions, f and g, is one-to-one
(b) Express in terms of .
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,
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 .