Total Function Spaces

Total Function
A function f is total on S if every element of S has an output. That is, dom(f) = S.
If every input in S is used, the function is total on S.
Total Function Space
The total function space S −tot→ T is the set of all functions f such that: - f ∈ S → T - dom(f) = S
Only functions that are defined for every element in S are included.
Example: Total Functions
Using earlier examples:
f ∈ {0,1,2,3,4} −tot→ N
g ∈ N −tot→ N
Small Total Function Space Example
A small complete function space can be fully listed.
{0,1} −tot→ {2,3} contains exactly 4 functions:

{(0,2),(1,2)}
{(0,2),(1,3)}
{(0,3),(1,2)}
{(0,3),(1,3)}
Key Idea
Total functions are defined for every input in the domain set S.

Quiz (Instant Check)

1. What does it mean for a function to be total?



2. What is dom(f) for a total function on S?



3. What is S −tot→ T?



4. How many functions are in {0,1} −tot→ {2,3}?



5. What makes a function NOT total?