READING: I §3 6
Back to reading:
I §3 5
begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] §3. FUNCTIONS. "end quote
These pages assume that the reader has the book and is reading along. topic index contents
Please read these notes from the beginning to here and then paragraph 6 of §3.
COMPOSITION OF FUNCTIONS
If f : X → Y and g : Y → Z are functions, one can define the composition of f and g, or composed function, a function from X into Z by associating to each element of X and element of Z in the obvious way: given an element of X, one first uses f to get an element of Y, then one uses g to get from this last element an element of Z. The composed function is usually denoted g f, so that we have g f : X → Z, with ( g f )(x) = g( f (x)) for each .
This composition of functions is a very simple idea, but I find that it is also confusing. We mention the function f before the function g and we call it the composition of f and g, but the symbol g f has g first. This last fact, however, is a memory device for the composition's value: ( g f )(x) = g( f (x)). The parentheses in this expression will also be a source of confusion for those who are not familiar with standard mathematical notation. The first set of parentheses indicate that g f is the symbol or name of the composed function, just as f and g are the names or symbols of functions. All three other pairs of parentheses enclose the arguments of the functions. Finaly, I would like to look at problem 6 on page 12:
Prove that if f : X → Y, g : Y → Z and h : Z → W are functions, then h ( g f ) = ( hg ) f.
Let the proof be an exercise, and please see the example here.