11. OCTOBER 6TH: FUNCTIONS: EXAMPLES. COMPOSITION OF FUNCTIONS.
27
11. October 6th: Functions: examples. Composition of functions.
11.1
.
Let
A
denote a set
{
F, T
}
with exactly two elements, and let
S
be any set.
Prove that giving a function
S
→
A
(that is, an element of
A
S
) is ‘the same as’ giving
a subset of
S
(that is, an element of
P
(
S
)). In other words, explain how to define a
subset of
S
for every function
S
→
{
F, T
}
, and how to define a function
S
→
{
F, T
}
for every subset of
S
, in compatible ways.
(This is why
P
(
S
) is also denoted 2
S
;
here, ‘2’ stands for a set with two elements.)
Answer:
Given a function
f
:
S
→
{
F, T
}
, let
X
f
⊆
S
be the subset defined by
X
f
=
f
−
1
(
T
) =
{
s
∈
S

f
(
s
) =
T
}
.
In this way, every function
S
→
{
F, T
}
determines a subset of
S
.
Conversely, given a subset
X
⊆
S
of
S
, define a function
f
X
:
S
→
{
F, T
}
by
prescribing
∀
s
∈
S
f
X
(
s
) =
T
if
s
∈
X
F
if
s
∈
X
.
In this way, every subset of
S
determines a function
S
→
{
F, T
}
.
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 Fall '11
 Aluffi
 Set Theory, Existence, codomain, Function composition

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