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.