7 Funkcje,relacje i porządki.pdf

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litm.dvi
7
7 A
7 P
7 C
7 F
7 H
7 J
7 O
69697337.007.png 69697337.008.png
*
*
u×v
R S T
*
D(R) d = {x:∃ y (x, y)∈R} D (R) d = {y:∃ x (x, y)∈R}
< ≤
*
R
R⊆D(R)×D (R)
x∈ S S R y∈ S S R
*
∃ y (x, y)∈R ∃ x (x, y)∈R
xRy
xRy
x
R
y
∈ u
x∈ u y , x∈u ^ y∈u ^ (x∈y _ x = y)
⊆ u
x⊆ u y , x∈u ^ y∈u ^ x⊆y
= u
x = u y , x∈u ^ x = y
7
7 A
(x, y)∈R
69697337.009.png 69697337.010.png
*
R
R −1
= {(y, x): (x, y)∈R}
R S
RS xRSy ,∃ z (xRz ^ zSy)
u
u
∀ x2u (xRx)
u
∀ x2u ∀ y2u ∀ z2u ((xRy ^ yRz) ) xRz)
u
∀ x2u ∀ y2u (xRy _ yRx)
*
= u
*
⊆ u
u ∈ u
7
7 B
*
R
u
∀ x2u ¬(xRx)
∀ x2u ∀ y2u (xRy ) yRx)
u
∀ x2u ∀ y2u ((xRy ^ yRx) ) y = x)
69697337.001.png 69697337.002.png
*
R
x
y
x
= u
R y
*
u×{v}
e
f
g
h
f
y
*
xfy
f : u ! v u∋x ! f(x)∈v
f(x) f x
f
u
v
*
f : u ! v
f
u v f
u v
u v
v u v u
v u = {f∈P(u×v) : f : u ! v}
7
7 C
69697337.003.png 69697337.004.png
*
f
u⊆D(f)
f| u
= {(x, y)∈f : x∈u}
f[u] d = D (f| u )
f[u]
{f x } x2u
u f
*
u⊆D(f) v⊆D(f)
f[u] \ f[v]⊆f[u \ v]
f[ S u] = S {f[x] : x∈u}
f[ T u]⊆ T {f[x] : x∈u}
u⊆v⊆D(f)
f[u]⊆f[v]
*
f
u⊆D(f)
S
x2u f x
= S f[u] T
x2u f x
= T f[u]
7
7 D
{f(x): x∈u}
x⊆D(f)
x∈u
x⊆D(f)
x∈u u =∅
69697337.005.png 69697337.006.png

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