Functions
| Lecture #2: Review of Relevant Mathematics |
Functions  | Informal notion of functions informally, a function associates one "output" with each of several possible "inputs"
the set of possible inputs is called the function's domain, the possible outputs its range |
| Lecture #2: Review of Relevant Mathematics |
Functions  | Informal notion of functions |
| | Functions as (restricted) sets of pairs formally, in set theory, functions are sets of pairs such that no "left half" values from the domain are duplicated (i.e., so that the association between domain and range values is unique) |
| Lecture #2: Review of Relevant Mathematics |
Functions | | Informal notion of functions |
| | Functions as (restricted) sets of pairs |
| | The function space operator the set of all functions from a set A to a set B is called the function space A ® B
Def: A ® B = { f Í A ´ B | " a Î A $ a unique b such that (a,b) Î f } |
| Lecture #2: Review of Relevant Mathematics |
Functions | | Informal notion of functions |
| | Functions as (restricted) sets of pairs |
| | The function space operator |
| | Cardinality issues the cardinality of the function space A ® B is the exponent | B | | A | |
| Lecture #2: Review of Relevant Mathematics |
Functions | | Informal notion of functions |
| | Functions as (restricted) sets of pairs |
| | The function space operator |
| | Cardinality issues |
| | Functions as realized by computations in Computer Science we generally think of functions as computed by algorithms, programs, etc. The tension between this more concrete sense and the more abstract mathematical sense is one of the key themes of this course |
| Lecture #2: Review of Relevant Mathematics |
Functions | | Informal notion of functions |
| | Functions as (restricted) sets of pairs |
| | The function space operator |
| | Cardinality issues |
| | Functions as realized by computations |
| | Partial functions vs. total functions if a function is not defined for some elements of its "natural" domain, we say that it is a partial function;
in cases involving computation, functions are often partial because the associated computation does not terminate |
| Lecture #2: Review of Relevant Mathematics |
Functions | | Informal notion of functions |
| | Functions as (restricted) sets of pairs |
| | The function space operator |
| | Cardinality issues |
| | Functions as realized by computations |
| | Partial functions vs. total functions |
| | Relations as functions to booleans another view of relations (and one taken by the textbook) is that a relation over sets A and B is just a function from A ´ B to the set of boolean values
Q: does this view respect our rules for cardinalities? |