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The Mathematics of
Functions
The
λ
calculus is very low level. Here we investigate functions in a more abstract setting.
This is very similar to writing algorithms in English prose rather than writing actual
machine code for the Turing machines.
–p
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Functions
Function Application and Composition:
Let
f
:
A
→
B
be a (total) function from
A
to
B
, then for every value
x
∈
A
we obtain a value
y
∈
B
,
fx
=
y
Function application is expressed by the
juxtaposition
of the function and its argument and is
evaluated from right to left.
Now assume that we have another function
g
:
B
→
C
from
B
into
C
, then we can apply the
function
g
to the result of
f
. For every value in
x
∈
A
we obtain a value
z
∈
C
,
gfx
=
gy
=
z
In other words, we just constructed a new function, call it
h
:
A
→
C
, such that
hx
=
gfx
=
gy
=
z
We can express the same idea using function composition,
◦
, without having to explicitly reference
any values in
A
,
B
,or
C
,
h
=
g
◦
f
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This note was uploaded on 10/03/2011 for the course CSC 544 taught by Professor Staff during the Spring '11 term at Rhode Island.
 Spring '11
 Staff
 Algorithms

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