A function is a functional and total relation.
1.1.1 Functions
Throughout this work, we will sometimes denote a function
- 1.
For example, given a function
taking values on a set of functions such as
, we will sometimes also write - 2.
This notational choice is based on the lambda notation
but uses a “
” symbol for better spacing and double brackets instead of either:- (a)
Square brackets
; - (b)
Parentheses
;
hoping to improve readability when dealing with e.g.:
- (c)
Equivalence classes, cf.:
- (i)
- (ii)
- (iii)
- (i)
- (d)
Function evaluations, cf.:
- (i)
- (ii)
- (iii)
- (i)
- (a)
Square brackets
- 3.
We will also sometimes write
, , etc. for the arguments of a function. Some examples include:- (a)
Writing
for a function . - (b)
Writing
for a function . - (c)
Given a function
, writingfor the function
. - (d)
Denoting a composition of the form
by
.
- (a)
Writing
- 4.
Finally, given a function
, we writefor the value of
at some .
For an example of the above notations being used in practice, see the proof of the adjunction
stated in Chapter 2: Constructions With Sets, Item 2 of Proposition 2.1.3.1.3.