# functions proof (is it right?)

• May 10th 2009, 01:42 PM
pberardi
functions proof (is it right?)
Prove that if
g: A -> B is 1-1
f: B -> C is 1-1
then fog: A -> C is 1-1

pf.
Need to show that if f(g(a)) = f(g(c)) then a = c for fog to be 1-1

Because g is 1-1, g(a) = g(b) and a = b
Because f is 1-1, f(b) = f(c) and b = c therefore a = c
Therefore since a = c, fog is a 1-1 function.
• May 10th 2009, 01:56 PM
Plato
Quote:

Originally Posted by pberardi
Prove that if
g: A -> B is 1-1
f: B -> C is 1-1
then fog: A -> C is 1-1

pf.
Need to show that if f(g(a)) = f(g(c)) then a = c for fog to be 1-1

Because g is 1-1, g(a) = g(b) and a = b
Because f is 1-1, f(b) = f(c) and b = c therefore a = c
Therefore since a = c, fog is a 1-1 function.

It should be:
$\begin{gathered}
f \circ g(a) = f \circ g(a) \hfill \\
g(a) = g(a)\;,\;f~\text{is one-to-one} \hfill \\
a = b\;,\;g ~\text{is one-to-one}\hfill \\
\end{gathered}$