关键词 > ECON6003
ECON6003W1
发布时间:2022-04-12
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ECON6003W1
1. Proposition:
a) Definition:a
statement
that
is
unambiguously
either
true
or
false
(but
not both)
in
a
given
context
b) Logicaloperation
i. not: ¬
ii. and: ∧
iii. or: ∨ (inclusivein
mathematics)
* exclusive“or”
in
math:
p
or
q
but
not
both
iv. ifp,
then
q:
⇒
;
p
if
and
only
if
(iff)
q:
⟺
* “ifp
is
F,
then
⇒
” is
vacuously
true.
* ⇒
is
true
except
if
p
is
true
and
q
is
false.
Ex.“1 + 4 = 9 ⇒ 8 < 1”
is
vacuously
true;
“1 + 2 = 3 ⇒ 8 < 1 ”is
false; “1 + 1 = 3 ⇒ 8 > 1”
is
vacuously
true.
v. ¬( ⇒
): ¬
⇒ ¬
; ¬(
⟺
): ¬
⟺ ¬
vi. DeMorgan’slaw: ¬(
∧
) ≡ ¬
∨ ¬
; ¬(
∨
) ≡ ¬
∧ ¬
|
|
|
|
¬ |
|
|
T
|
T
|
T |
T |
F |
T |
T |
T
|
F
|
F |
T |
T |
F |
F |
F
|
T
|
F |
T |
T |
T |
F |
F |
F |
F |
F |
T |
T |
T |
2. Proof ofa
proposition:
assumption
+
logical
operation
a) Constructive/deductiveproof
b) Contraposition:conversion
of
a
proposition
from
all ¬
⇒ ¬
to all
⇒
c) Bycontradiction
(most
common
one) from
all
⇒ ¬
is
false
to all
⇒
-
d) Byinduction
3. Set:a well-specific collection
of distinct objects
which
are
called
“elements”
a) Collection:no
sequence
while
= (
,
,
) has
sequence.
b) Distinct:{a,
a,
b,
c}
is
not
a
valid
set
c) Description:
i. Bylisting
of
all
its
elements:
= {
,
,
}
ii. Bydescribing
elements’
common
property:
= {
∈ ℝ|
≥ 0} or
= {
|
= 10 ∗
,
= 1,2,3,4} * The
property
should
be
a
statement
that
is
true
or
false. (it’s
not ambiguous)
d) Well-specific: = {
∈ ℝ |
≥ 0};
= {
|
= 10 ∗
,
=
,
,
, }
e) Symbol
i. ∈
: a
is
an
element
of
set
A;
a belongs
to
A
ii. ∉
: d
is
not
in
A
iii. ⊆
: A
is
a
subset
of
B;
every
element
of
A
is
in
B
(
∈
⇒
∈
);
* ⊆
⊆
⇒
=
f) Empty/nullset ∅:
the
unique
set
having
no
element
* ∅ isthe
subset
of
any
non-empty
set
(trivially
or
vacuously
true) The definition
of
subset
is
⊆
∈
⇒
∈
.
Then,
if
= ∅ , then there
is
no
∈
, which
means “
∈
” if false. According
to “if
p
is
F,
then
⇒
”is
vacuously
true, “∅ is
the
subset
of
any
non-empty
set”isvacuously true.
g) Universalset:
collection
of
all
the
elements
under
consideration.
E.g.
price
4. Operation:(given
,
⊆
)
a) Completement ! = {
∈
|
∉
}
b) Union ∪
= {
∈
|
∈
∈
}
c) Intersection ∩
= {
∈
|
∈
∈
}
d) Setminus
\
= {
∈
|
∈
∉
} =
∩
!
e) Symmetricdifference
∆
= (
\
) ∪ (
\
) = (
∪
)\(
∩
)
-
f) CartesianProduct
×
= {(
",
# )|
" ∈
,
# ∈
} (ex.)4.
* orderedpair (
",
# )
* usuallydifferent
from
×