Download - Semântica Proposicional Categórica
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→
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SE T →
SE T →
SE T →
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SE T →
SE T
L
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L
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→
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¬A = A → ⊥ ¬A ⊥
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∀ ∃
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C
C
C
C
D
CD
f : a −→ b f : a, b
f
a −→ b D(f ) = a CD(f ) = b
◦ f : a −→ b
g : b −→ c h : c −→ d (h ◦ g) ◦ f = h ◦ (g ◦ f ) :
a −→ d
a f
g◦f
b
g
h◦g
c
h d ;
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id
a
b
id : Obj(C ) −→ M or(C )
ida : a −→ a idb : b −→ b
f : a −→ b
a f
f
b
idb
a f b
b a
ida
f
idb ◦ f = f ◦ ida = f
SE T →
C Obj(C ), Mor(C ), D, CD, id, ◦
SE T →
SE T
Obj(SE T )
Mor(SE T )
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Mor(SET ) −→ Obj(SE T )
D(f ) = A
CD(f ) = B
f : A −→ B
◦ : M or(SE T )2 −→ M or(SE T )
id : Obj(SE T ) −→ M or(SE T )
idA : A −→ A
idB : B −→ B
◦
f : A −→ B g : B −→ C h : C −→ D
a ∈ A
((h ◦ g) ◦ f )(a) = (h ◦ g)(f (a)) = h(g(f (a))) = h(g ◦ f (a)) = (h ◦ (g ◦ f )))(a) .
f : A −→ B a ∈ A b ∈ B
f (a) = b
(f ◦ idA)(a) = f (idA(a)) = f (a) = b = idB(b) = idB(f (a)) = (idB ◦ f )(a) .
SE T
= {0, 1, 2, 3, . . .}
SE T
m
n
n + m
n ◦ m = n + m
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n
n+m
m
m + n
m + (n + k) = (m + n) + k
n
m
k
n
m◦n
m
k◦m
k
m ◦ (n ◦ k) = (m ◦ n) ◦ k = m + (n + k) = (m + n) + k
id
N
0
m
m
0
n
n
0 + m = m n + 0 = n
→
SE T →
Obj(SE T →)
Mor(SE T →)
h, k
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A h
f
C
g
B
k D ,
g ◦ h = k ◦ f
Mor(SET →) −→ Obj(SE T →)
f : i, j −→ h, k D(f ) = i, j CD(f ) = h, k
◦ : Mor(SE T →)2 −→ Mor(SE T →)
Mor(SET →)
i, j ◦ h, k i ◦ h, j ◦ k
id : Obj(SET →) −→ M or(SE T →) idA, idB
◦ SE T →
A h
f
C i
g
N
l
B k D j M ,
i, j h, k i, j ◦ h, k
A i◦h
f
N
l
B j◦k M .
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i, j ◦ h, k = i ◦ h, j ◦ k .
(i, j ◦ h, k) ◦ f, g = (i ◦ h) ◦ f, ( j ◦ k) ◦ g =
i ◦ (h ◦ f ), j ◦ (k ◦ g) = i, j ◦ h ◦ f, k ◦ g =
i, j ◦ (h, k ◦ f, g)
SET →
A idA
f
A
f
B
idB
B ,
A
f −→ B idf = idA, idB
A idA
f
A h
f
C
g
B
idB B
k D ,
h, k ◦ idA, idB = h ◦ idA, k ◦ idB = h, k
idA, idB ◦ h, k = idA ◦ h,idB ◦ k = h, k
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POSET
Obj(POSET )
Mor(POSET )
{a, b} ∈ a ≤ b f : A −→
B
f (a) ≤ f (b)
Mor(POSET ) −→ Obj(POSET ) f :
A −→ B D(f ) = A CD(f ) = B
◦ : M or(POSET )2 −→ Mor(POSET )
id : Obj(POSET ) −→ Mor(POSET )
idA : A −→ A
idB : B −→ B
◦
M, ≤
N, ≤ f (≤) : M, ≤ −→ N, ≤
f (≤) : M −→ N ≤
(m, n) ≤
f ◦ ≤ = ≤ ◦ f
f
m ≤ n
f
(f (m), f (n))
≤ f (m ≤ n) = f (m) ≤ f (n)
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f (≤) : M, ≤ −→ N, ≤
f (m ≤ n) = f (m) ≤ f (n)
f : A −→ B g : B −→ C h : C −→ D
a ≤ b A f (a) ≤ f (b) B f : A −→ B
f (a) ≤ f (b) f (a) ≤ f (b) = f (a ≤ b)
g
h
((h ◦ g) ◦ f )(a ≤ b) = (h ◦ g)(f (a ≤ b)) = h(g(f (a ≤ b))) = h(g ◦ f (a ≤ b)) =
(h ◦ (g ◦ f )))(a ≤ b) .
f : A −→ B {a, b} ∈ A a ≤ b
f (a) ≤
f (b)
f (a ≤
b) = f (a) ≤
f (b)
f ◦ idA(a ≤ b) = f (idA(a ≤ b) = f (a ≤ b) = f (a) ≤ f (b) = idB(f (a) ≤ f (b)) =
idB(f (a ≤ b) = idB ◦ f (a ≤ b)
M, ⊕, e
M, ⊕
⊕ : A × A −→ A
{a,b,c} ∈ A
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a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c
e ∈ M ⊕
a ∈ M
a ⊕ e = e ⊕ a = a
{a, b} ∈ M a ⊕ b = b ⊕ a
, +
, · 0 1
, +, 0 , ·, 1
M, ⊕, a N,,e {∗}
f (⊕) : M, ⊕, a −→
N,,e f (⊕) : M −→ N
M
M
M
Mor(M)
◦ : M ×M −→ M idM
M f ,g,h ∈ Obj(M) f ◦(g◦h) = (f ◦g)◦h = f ⊕(g⊕h) = (f ⊕g)⊕h idM M idM = e
f ◦ idM = idM ◦ f = f = f ⊕ e = e ⊕ f
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(m, n) ⊕
f ◦ ⊕ = ◦ f
f
m ⊕ n
f
(f (m), f (n))
f (m ⊕ n) = f (m) f (n)
f (⊕) : M, ⊕, e −→ N,,d
f (m ⊕ n) = f (m) f (n)
{∗}a
e
M f
N
f : {a} −→ {e}
f (a) = e
f ◦ ⊕ = ◦ f f (⊕) = (f )
m n
f (⊕(m, n)) = (f (m, n))
f (⊕(m, n)) = (f (m), f (n)) f (m ⊕ n) = f (m) f (n)
, +, 0 A, ×, 1 A =
{2n | n ∈ } f : −→ A f (x) = 2x
, +, 0 A, ×, 1
0
1
f (0) = 20 = 1
a
b
f (a+b) =
2a+b
f (a + b) = 2a+b = 2a × 2b = f (a) × f (b)
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MON
Obj(MON )
M, ⊕, e N,, d
Mor(MON )
h : M, ⊕, e −→ N,, d
h : M −→ N
Mor(MON ) −→ Obj(MON ) f : M −→ N
D(f ) = M, ⊕, e
CD(f ) = N,, d
f : M, ⊕, e −→ N,, d
◦ : M or(MON → M ON )2 −→ M or(MON → M ON )
id : Obj(MON ) −→ M or(MON )
idM : M, ⊕, e −→ M, ⊕, e
idN : N,, d −→ N,, d
◦
M f
g◦f
N
g
h◦g
G
h H
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M, ⊕, a N,,e G, , o
H, , u
{m, n} ∈ M f : M −→ N f (m ⊕ n) =
f (m) f (n) f (a) = e
{ p, q } ∈ N g : N −→ G g( p q ) = g( p) g(q )
g(e) = o
{r, s} ∈ G
h : G −→ H
h(r s) = h(r) h(s)
h(o) = u
{m, n} ∈ M g ◦ f : M −→ G g ◦ f (m ⊕ n) =
g ◦ f (m) g ◦ f (n) g ◦ f (a) = o
{ p, q } ∈ N h ◦ g : N −→ H h ◦ g( p q ) =
h ◦ g( p) h ◦ g(q ) h ◦ g(e) = u
{m, n} ∈ M (h ◦ g) ◦ f = h ◦ (g ◦ f ) : M −→ H
h ◦ (g ◦ f ) (m ⊕ n) = h ◦ (g ◦ f )(m) h ◦ (g ◦ f )(n) h ◦ (g ◦ f )(a) = u
=⇒ (h ◦ g) ◦ f (a) = h ◦ (g ◦ f )(a)
(h ◦ g) ◦ f (a) = h(o)
(h ◦ g)(e) = h(o) h(o) = u
(h ◦ g)(e) = u (h ◦ g) ◦ f = h ◦ (g ◦ f )
⇐= h◦(g◦f )(a) = u h◦(g◦f )(a) = h◦g(e)
h◦(g◦f )(a) = (h◦g)◦f (a)
(h ◦ g) ◦ f = h ◦ (g ◦ f )
f : M −→ N
a ∈ M
e ∈ N
f (a) = e
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(f ◦ idM )(a) = f (idM (a)) = f (a) = e = idN (e) = idN (f (e)) = (idN ◦ f )(a) .
G, , e,−1
G,
: A × A −→ A
{a,b,c} ∈ A
a (b c) = (a b) c
e ∈ G
a ∈ G
a e = e a = a
e ∈ G a−1 ∈ G
a ∈ G
a a−1 = a−1 a = e
a, b ∈ G a b = b a
, + , ·
0
1
−a ∈
1a
∈ a = 0
, +, −a
, ·, 1a
a a a a = a
a (a a) = (a a) a = e a = a
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G, , e,−1
H,, d,−1
{∗}
G
H
g() : G, , e,−1 −→
H,, d,−1 g() : G −→ H
G
(a, b)
∗
g ◦ = ◦ g
g
a b
g
(g(a), g(b))
g(a b) = g(a) g(b)
g() : G, , e −→ H, , d
g(a b) = g(a) g(b)
{∗}e
d
G g H
g : {e} −→ {d}
g(e) = d
g(a) g−1 g(g(a))−1
a
g
g−1◦ g
g(a−1) = g(g(a))−1
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a a−1 = e a−1 a = e
g(a a−1) = g(e)
g(a) g(a−1) =
g(a−1) g(a) = g(e) = d g(a−1) = (g(a))−1
GRU
Obj(GRU )
G, , e,−1 H,,d,−1
Mor(GRU )
h : G, , e,−1 −→ H,,
d,−1 h : G −→ H
M or(GRU ) −→ Obj(GRU ) f : G −→ H D(f ) =
G, , e,−1 CD(f ) = H,, d,−1 f
f : G, , e,−1 −→ H,, d,−1
◦ : M or(GRU → GRU )2 −→ M or(GRU → GRU )
id : Obj(GRU ) −→ Mor(GRU )
idG : G, , e,−1 −→
G, , e,−1 idH : H,, d,−1 −→ H,, d,−1
◦
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M f
g◦f
N
g
h◦g
G
h H
M, ⊕, a,−1 N, , e ,−1
G, , o,−1 H, , u,−1
{m, n} ∈ M f : M −→ N f (m ⊕ n) =
f (m) f (n) f (a) = e
f (m−1) = f (f (m))−1
{ p, q } ∈ N g : N −→ G g( p q ) = g( p) g(q )
g(e) = o
g( p−1) = g(g( p))−1
{r, s} ∈ G h : G −→ H h(r s) = h(r) h(s)
h(o) = u, e h(r−1) = h(h(r))−1
{m, n} ∈ M g ◦ f : M −→ G g ◦ f (m ⊕ n) =
g ◦ f (m) g ◦ f (n) g ◦ f (a) = o g ◦ f (m−1) = g ◦ f (g ◦ f (m))−1
{ p, q } ∈ N h ◦ g : N −→ H h ◦ g( p q ) =
h ◦ g( p) h ◦ g(q ) h ◦ g(e) = u h ◦ g( p−1) = h ◦ g(h ◦ g(m))−1
{m, n} ∈ M (h ◦ g) ◦ f = h ◦ (g ◦ f ) : M −→ H
h ◦ (g ◦ f ) (m ⊕ n) = h ◦ (g ◦ f )(m) h ◦ (g ◦ f )(n) h ◦ (g ◦ f )(a) = u
h ◦ (g ◦ f )(m−1) = h ◦ (g ◦ f )(h ◦ (g ◦ f )(m))−1
M, ⊕, a,−1 m⊕ m−1 = a
f (m⊕ m−1) = f (a)
f (a) = e
(1)
f (m ⊕ m−1) = e
(5) h ◦ g(e) = u h ◦ g(f (m ⊕ m−1)) = (h ◦ g) ◦ f (m ⊕ m−1)) = u
u = h ◦(g ◦f )(a) m⊕ m−1 = a u = h ◦(g ◦f )(m⊕ m−1)
(h ◦ g) ◦ f (m ⊕ m−1) = h ◦ (g ◦ f )(m ⊕ m−1)
(h ◦ g) ◦ f = h ◦ (g ◦ f )
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f : M −→ N
a ∈ M e ∈ N f (m ⊕ m−1) = f (a) = e
(f ◦ idM )(a) = f (idM (a)) = f (a) = e = idN (e) = idN (f (e)) = (idN ◦ f )(a) .
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f : A −→ B x1 x1 ∈ A
x2 x2 ∈ A x1 = x2 f (x1) = f (x2)
f : A −→ B
g, h : C
A f ◦ g = f ◦ h g = h
f
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A f B
C
h
g
f ◦h=f ◦g
C h
g
A
f
Af
B ,
=⇒ f : A −→ B g, h : C A
f ◦ g = f ◦ h g = h
x ∈ C g h C f ◦ g = f ◦ h
(f ◦ g)(x) = (f ◦ h)(x) f (g(x)) = f (h(x)) f
g(x) = h(x)
g = h
⇐= {x1, x2} ∈ A f (x1) = f (x2) x1 = x2
g
h
C
g, h : C
A
h
g
z 1
z 2
C
A
g(z 1) = x1 h(z 2) = x2 {x1, x2} ∈ A
f ◦ g = f ◦ h g h
C
f (g(z 1)) = f (h(z 2)) f
g(z 1) = h(z 2) x1 = x2
f : A −→ B C
g, h : C A f ◦ g = f ◦ h g = h
f : A B
SE T
m + n = m + p
n = p
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f : A −→ B y
y ∈ B x x ∈ A f (x) = y
f : A −→ B
g, h : B C g ◦ f = h ◦ f g = h f
A f
g◦f =h◦f
B
g
h
C
A f
f
B
g
B
h C ,
=⇒ f : A −→ B g, h : B C
g ◦ f = h ◦ f g = h
x ∈ A g ◦ f = h ◦ f g(f (x)) = h(f (x)) y ∈ B f
x ∈ A f (x) = y g(f (x)) = h(f (x))
g(y) = h(y)
g = h
⇐= g ◦ f = h ◦ f x ∈ A y ∈ B f (x) = y
g = h
y ∈ B f x ∈ A f (x) = y
g(y) = g(f (x)) = (g ◦ f )(x) (g ◦ f )(x) = (h ◦ f )(x) = h(f (x)) = h(y)
g(y) = h(y)
g = h
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f : A −→ B C
g, h : B
C g ◦ f = h ◦ f g = h
f
f : A B
SE T
n + m = p + m
n = p
f : A −→ B
y
y ∈ B x x ∈ A f (x) = y
x1 = x2 f (x1) = f (x2) {x1, x2} ∈ A
g = g
g
g ◦ f = idA f ◦ g = idB
y ∈ B y = f (x) g(y) = g(f (x)) = (g ◦ f )(x) =
idA(x) = x g(y) = g(f (x)) = (g ◦ f )(x) = idA(x) = x g = g
g = idA ◦ g
= (g ◦ f ) ◦ g = g ◦ (f ◦ g ) = g ◦ idB = g g
f
f −1
f −1
f −1 ◦ f = idA
f ◦ f −1 = idB
f : A −→ B
g : B −→ A f ◦ g = idB g ◦ f = idA
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A
id
f
B
g
id
A f
idA
B
g
A f B
A B
g
idB
g ◦ f = idA f ◦ g = idB ,
=⇒ f : A −→ B
x
x ∈ A y ∈ B f (x) = y
y
y ∈ B
x ∈ A g(y) = x
g(y) = x
f (x) = y
y ∈ B (f ◦ g)(y) = f (g(y)) = f (x) = y = idB(y) f ◦ g = idB
(g ◦ f )(x) = g(f (x)) = g(y) = x = idA g ◦ f = idA
⇐= f f
f
f ◦ g = f ◦ h g = h
f −1 ◦ (f ◦ g) = f −1 ◦ (f ◦ h) f −1
(f −1 ◦f )◦g = (f −1 ◦f )◦h
idA ◦ g = idA ◦ h
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f
g ◦ f = h ◦ f g = h
(g ◦ f ) ◦ f −1 = (h ◦ f ) ◦ f −1 f −1
g ◦(f ◦f −1) = h ◦(f ◦f −1)
g ◦ idB = h ◦ idB
f
f : A −→ B C
f −1 : B −→ A
f −1 ◦ f = idA f ◦ f −1 = idB
f : A ∼= B f : A B
C 0 Obj(C ) 0
C a ∈ Obj(C )
!0 : 0 −→ a
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A
A =
⇐= A = −→ B
=⇒ A = B
B = {x,y,...} x = y f : A −→ B
A
A = {x1,...} f, g : A −→ B f (z ) = x
g(z ) = y
z ∈ A f = g x = y A
B B , , , −, 0, 1
,
−
0
1
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{x,y,z } ∈
AxB {x, y} ∈
x y = y x
AxB {x, y} ∈
x y = y x
AxB {x,y,z } ∈
x (y z ) = (x y) (x z )
AxB {x,y,z } ∈
x (y z ) = (x y) (x z )
AxB {x, 0} ∈
x 0 = 0 x =