Logic
Set Theory
Quantification
Unified
Logic is Logic:
Boolean Truth Values
Set Theory is Set Theory:
Manipulation of Sets
Quantification is a way to tie Logic and Set Theory together.
Special Case:
The Empty Set
∅
The Empty Set
∅
contains no elements.
Because it is a Set, you can Quantify over it.
∃
x
∈∅
:
P
(
x
)≡
FALSE
P(x) is arbitrary (it doesn't matter what the predicate is).
There are no elements in
∅
so no
x
∈∅
can exist.
∃
x
∈∅
:
P
(
x
)≡
FALSE
∃
x
∈∅
:
P
(
x
)≡
NOT
(∀
x
∈∅
:
NOT P
(
x
))
∀
x
∈∅
:
NOT P
(
x
)≡
NOT
(∃
x
∈∅
:
P
(
x
))
∀
x
∈∅
:
NOT P
(
x
)≡
NOT
(
FALSE
)
NOT
(
FALSE
)≡
TRUE
∀
x
∈∅
:
NOT P
(
x
)≡
TRUE
P(x) is arbitrary (it doesn't matter what the predicate is).
If P(x) is arbitrary so is NOT P(x).
Two rules:
∃
x
∈∅
:
P
(
x
)≡
FALSE
∀
x
∈∅
:
P
(
x
)≡
TRUE
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Implication
(Logic)
Implication is generally used when you must Quantify over a set (tying Logic and Set Theory together).
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 Spring '11
 Albert
 Logic, Tuple, Logical connective

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