Functions

• October 15th 2008, 12:06 PM
Aryth
Functions
Find a function $f: D \to E$ such that $f[f^{-1}[E]] \neq E$ where $f[E]$ is the image of f at E and $f^{-1}[E]$ is the pre-image of f at E.
• October 15th 2008, 03:34 PM
Plato
Suppose that $D = \left\{ {1,2,3} \right\}\,\& \,E = \left\{ {a,b,c} \right\}$.
Define a function
$\begin{gathered} f = \left\{ {\left( {1,a} \right),\left( {2,c} \right),\left( {3,a} \right)} \right\} \hfill \\
f^{ - 1} (E) = D \hfill \\ f\left( {f^{ - 1} (E)} \right) = f\left( D \right) = \left\{ {a,c} \right\} \ne E \hfill \\ \end{gathered}$
• October 17th 2008, 08:22 AM
poutsos.B
Quote:

Originally Posted by Aryth
Find a function $f: D \to E$ such that $f[f^{-1}[E]] \neq E$ where $f[E]$ is the image of f at E and $f^{-1}[E]$ is the pre-image of f at E.

The theorem covering the above is:

For all functions f : ,if f is not onto then f[ $f^-1$ ( R(f))]=/= R(f) ..........where R(f) is the range of f.

But due to contrapositive law the above theorem is equivalent to:

For all functions f : if f[ $f^-1$ ( R(f))]= R(f) then f is onto.

So any function which is not onto will satisfy f[ $f^-1$ ( R(f))]=/= R(f).

Now to prove the above .

But to prove that f is onto if f[ $f^-1$ ( R(f))]= R(f),we must prove that:

For all yεR(f) then there exists an xεD(f) such that y=f(x),where D(f)is the domain of f .

The above in symbols is:

$\forall y$[ yεR(f) -------> $\exists x$( xεD(f) & y=f(x))].

Let yεR(f),since f[ $f^-1$ ( R(f))]= R(f),then yεf[ $f^-1$ ( R(f))].

But yεf[ $f^-1$ ( R(f))] <=====> y= f(x) & xε[ $f^-1$ ( R(f))]..

But xε[ $f^-1$ ( R(f))] <====> xεD(f) & f(x)εR(f).

Hence; yεf[ $f^-1$ ( R(f))]<=======> y=f(x) & xεD(f) & f(x)εR(f).

Thus we see that there an xεD(f) such that y= f(x), so f is onto