Sets (a logical
approach)
1
Sets and Logic
Early, we introduced the symbol
, a binary predicate,
where
means an object
(which
may
be a set) is
in
the set B.
We then
define
as follows:
e.g.,
proper subset (how would we define
this?)
∈
a
∈
B
a
⊆
A
⊆
B
↔ ∀
x
(
x
∈
A
→
x
∈
B
)
{1,8,4}
⊆
{8,4,1}
{1,8,4}
⊆
{8,9,4,2,1}
{1,8,4}
⊂
{8,9,4,2,1}
2
We then
define
using
logic
as follows.
A
⊈
B
~
∀
࠵?
( x
∈
A
x
∈
B)
∃
࠵?
~(x
∈
A
x
∈
B)
negate
∀
∃
࠵?
~(
x
A
∨
x
∈
B)
remove
→
∃
࠵?
~(x
A
∨
x
∈
B)
defined a new symbol
∃
࠵?
(x
∈
A
∧
x
∉
B)
de Morgan
We are getting pretty “loose here”. We aren’t numbering statements, we
are using logical equivalences to justify a proof step. We could even get rid
of a few of those symbols.
A
⊈
B iff ~
∀
࠵?
( if x
∈
A then x
∈
B)
iff
∃
࠵?
~( if x
∈
A then x
∈
B)
negate
∀
iff
∃
࠵?
~(if
x
A
∨
x
∈
B)
remove
→
iff
∃
࠵?
~(if x
A
∨
x
∈
B)
defined a new symbol
iff
∃
࠵?
such that
x
∈
A
∧
x
∉
B
de Morgan
⊈
↔
→
↔
→
↔
∼ ∈
↔
∉
↔
∼ ∈
∉
3
Sets remind us why we need a
universe
.
(Very over simplified history)
When set theory was axiomatized, there was an idea that
there were objects, and
collections
, or sets. Objects could
be elements of sets, but so could other sets.
The question arose – can a set contain itself? That is, is it
possible there exists a set
S
such that
࠵?
࠵?
࠵?
?
Then, some sets are “selfswallowing”, and some are “non
selfswallowing”.
4
That is, the set of all walruses contains only walruses, and
the set containing only Joan of Arc does not contain the set
containing Joan of Arc. Call these
runofthemill
sets.
Then we should have
S1
, the set of all selfswallowing
sets, and
S2
, the set of all runofthemill sets.
Clearly a set is either selfswallowing or runofthemill.
What about
S2
? Is it in
S1
or
S2
?
If S2 is runofthemill, it can’t be in S1. But then it should
be in S2, but then S2 is would not be the set of runofthe
mill sets because now it contains itself.
5
We could rewrite this using set builder notation and logic.
Let
be the set of runofthemill sets.
Let
where
S
can be any set. (note that there is a
implied in set builder notation).
Is
If so,
 by the definition of X  every element of
can’t contain
itself?
Suppose
. Then
. Why? Because X contains all the sets
such that don’t contain themselves and
.
Symbolically,
Either way, we end up with a contradiction, and our logical system
collapses. This is
Russell’s paradox,
which you can read about.
X
X
= {
S

S
∉
S
}
∀
X
∈
X
?
X
∉
X
X
X
∉
X
X
∈
X
X
∉
X
X
∈
X
↔
X
∉
X
.
6
Barber paradox: The barber shaves everyone who does not
shave himself. Then who shaves the barber?
This was used as a popular way of explaining the problem in
the set theory of Frege. There is a simple way to solve the
barber paradox, just rephrase it as “The barber shaves
everyone who does not shave himself, with the exception
that he shaves himself.”
To avoid the contradiction in set theory, we need to define a
universe U of objects, our universe of discourse, from which
other sets are constructed.
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 Spring '14