a) Show that a partial function from \(A\) to \(B\) can be viewed as a function \(f^{*}\) from \(A\) to \(B \cup\{u\}\), where \(u\) is not an element of \(B\) and
\(f^{*}(a)= \begin{cases}f(a) & \text { if } a \text { belongs to the domain } \\ & \text { of definition of } f \\ u & \text { if } f \text { is undefined at } a .\end{cases}\)
b) Using the construction in (a), find the function \(f^{*}\) corresponding to each partial function in Exercise 77.
Equation Transcription:
{
Text Transcription:
A
B
f
cup {u}
f(x)= {_u if f is undefined at a ^f(a) if a belongs to the domain of definition of f
Solution:
Step 1 ;
In this problem we have to prove that the function can be viewed as a function
where u is not an element of B and
This shows that is well defined.
For each it shows that either
is in the domain of the definition of
or it is not.
If is in the domain of the definition then
is the well defined element,
If is not in the domain of the definition
In either case is well defined.
