View source for FA18:Lecture 6 functions

We defined [[function]]s and give several examples of things that are [[function]]s
and are not [[function]]s.

* Last semester: [[SP18:Lecture 5 Functions|Lecture 5 Functions]] and [[SP18:Lecture 6 Function Properties|Lecture 6 Function Properties]]
* [[File:fa18-lec06-board.pdf]]

= Functions =

{{:Function}}

= Function examples and non-examples =

== Specifying functions ==

When giving a [[function]], you must indicate the [[domain]], the [[codomain]],
and an [[unambiguous]] rule giving an output for every input.  There are many,
many ways to do this, as long as the output is clear.  We gave a few examples
last lecture, here are some more.

=== direct specification ===

{{Example:specify a function directly}}

=== drawings ===

{{Example:drawing a function}}

=== tables ===

{{Example:table as a function}}

=== function description ===

{{Example:describing a function}}

=== lists / sequences ===

{{Example:list as a function}}

=== empty domain ===

{{:Claim:there is a unique function with empty domain}}

== Examples that are not functions ==

When we write down things that we claim are [[function]]s, we must check that
they are [[well-defined]], i.e. that they actually are functions.

=== ambiguous functions ===

{{Example:ambiguous function}}

=== undefined (partial) functions ===

{{Example:partial function}}

=== empty codomain ===

{{:Claim:There is no function with an empty codomain}}

= Operations on functions =

== Function equality ==

{{:Equality (functions)}}

== Image of a function ==

{{:image}}

== Function composition ==

{{:composition}}

== Identity function ==

The [[identity function]] is a simple but useful [[function]] (in fact,
[[Claim:the identity function is a bijection|it is a bijection]]):

{{:Identity function}}

== Set of all functions from X to Y ==

{{:⟦X→Y⟧}}