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L2.12
Lecture Notes: Set theory
A
set
is
a collection of objects from a universe.
The objects are called
members
or
elements
. They are not in any order,
despite how they may be written, nor may sets contain duplicate members.
Sets may be denoted in several ways:
(1). By enumeration, if finite. E.g. {1,2,3} (Same as {2,1,3} — same as
{1,1,2,3})
(2). By enumeration with dots of ellipses, where obvious. {2,4,6 . .}
(3). By use of a propositional function: {x  x>3} (Need to know universe!)
Or in general, {x  P(x)} or “the set of all elements that make P true.”
Here are some facts and definitions about sets: As always, there is U, the
universe of discourse, lurking in the background. Much of what we do with
sets involves logic, so the careful reader may want to review or have close at
hand our notes on logic.
Define: x
∈
S
to mean “x is an element of S.” Note that x
∈
S is a proposition.
We also define x
∉
S to mean
¬
(x
∈
S) or “x is not an element of S.”
Define S as “cardinality of S” or the number of elements in S.
Define: S
⊆
T as S is a subset of T iff
2200
x(x
∈
S
→
x
∈
T)
STOP: You should read the logic as smoothly as if it were written by J.K.
Rowling. If you cannot, go back and review logic!
Define: S=T
iff S and T contain the same elements, or iff
2200
x(x
∈
S
↔
x
∈
T)
Theorem: S=T iff S
⊆
T
∧
T
⊆
S (In words, two sets are equally if each is a
subset of the other.)
Proof: S=T iff
2200
x(x
∈
S
↔
x
∈
T)
iff
2200
x[(x
∈
S
→
x
∈
T)
∧
(x
∈
T
→
x
∈
S)]
iff
2200
x(x
∈
S
→
x
∈
T)
∧
2200
x(x
∈
T
→
x
∈
S)
iff S
⊆
T
∧
T
⊆
S
QED
Define:
∅
, or the
null
or
empty
set, as the set with no elements. That is, the
following is a true proposition:
2200
x(
¬
(x
∈∅
)) or
2200
x(x
∉∅
)
1
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View Full DocumentTheorem: for any set S,
∅⊆
S
Proof:
∅⊆
S iff
2200
x(x
∈∅
→
x
∈
S)
Since x
∈∅
is
false
for all x,
2200
x(x
∈∅
→
x
∈
S) is
true
and this is sufficient
to prove
∅⊆
S.
Theorem: for any set S, S
⊆
S.
Proof: S
⊆
S iff
2200
x(x
∈
S
→
x
∈
S)
Since P
→
P is a tautology the quantified statement is
true
, and the theorem
is proved.
Sets may be members of sets.
E.g. S = {1,2, {1,2}} is a set of cardinality 3,
containing two numbers and a set as members.
The most important set that contains sets as members is the
power set
of a
set S, denoted
P
(S), which is the
set of all subsets of S
.
E.g., Let S={a,b}.
P
(S) =
P
({a,b}) = {
∅
, {a}, {b}, {a,b}}
P
(
∅
) = {
∅
} (The set containing as its only member the null set)
P
(
P
(
∅
)) =
P
{
∅
} = {
∅
, {
∅
}}
P(P
(
P
(
∅
))) =
P
{
∅
, {
∅
}} = {
∅
, {
∅
}, {{
∅
}}, {
∅
, {
∅
}}}
STOP: Think about this!
What is
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 Spring '08
 WATKINS

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