Prednáška

Diskrétna matematika - Letný semester rok 2000

Def |C| = |A| + |B|

"!$#&%'()*,+.-/) "!02143&5$36#

1a) A1

∪B1 = C

1b) A1

∩B1 = ∅

1c)

|A|=|A1|, |B|=|B1| .

Def 7/8 |C|=|A|

|

B|, ak platí |C|=| AB| , (AB -

9:#,)!;=<>?@ ,<#&A' CB/A>6'A;=5ED>FBGA>H6'A;F1FICJ

Def K |C|=|A|*|B|, ak |C|=|AxB| .
Veta Nech |A|

L |X|, |B|L |Y| , potom platí

2a)

|A| + |B|

L |X| + |Y|,

2b)

|A| * |B|

L |X| * |Y|,

2c)

|A|

|

B|

L |X||Y|,

Veta

M 1@')B#,) 'N"!029O ( )A>( ) 'IQPR@#29:#,)!;FBGA>H6'A;F1/5STVUO ") W

3a)

|A| + |B| = |B| + |A| -

!>HB/*) ")XH9A>( (.Y&') >H9 "A'

3b) (

|A| + |B|) + |C| = |A| + (|B| + |C|) , -

(>N&' ")XH9A>( ("Y&') >H9 "A'

3c)

|A| * |B| = |B| * |A| - komutatývnos

AZ(.>?#,A'

3d) (

|A| * |B|) * |C| = |A| * (|B| * |C|) , -

(.>N&' ")XH9A>( AZ(>H?#,A'

3e)

|A| * (|B| + |C|) = (|A| * |B| + |A| * |C| )

Veta

M 1@')B#,) 'N"!029O ( )A>( ) 'IQPR@#29:#,)!;FBGA>H6'A;F1/5STVUO ") W

4a)

|A|

|

B| + |C| =|A||B| * |A||C|,

4b) (

|A| * |B|)

|

C| = |A||C| * |B||C|,

4c) (

|A|

|

B|)|C| = |A||B| * |C| .

Lema

PR@# *?>9> A-/BGA>H6'A*G[\UO ") W | P(X) | = 2 |X| .

Def |N|

(. ]><&A Y"*"+,#G^

0 - Alef nula .

Veta ℵ0 + ℵ0 = ℵ0 .

Veta ℵ0 * ℵ0 = ℵ0 .
Veta (Cantorova veta) |X| < | P(X) |= 2

|

X| .

Dôsledok 1. |X| < 2

|

X| .

Dôsledok 2.

_$##&%' ( )*,+,#2B/A>6'A C9:#,)!XN"`GBGA>6WAQJ

.RQHþQpPQRåLQ\

Def

abA>6'A C1c(. CA ,<X9 d He ak |A| < ℵ

0 -

Y&'6#S "! |A| < |N| .

Def Nn = {0, 1, 2, 3, ..., n-1} , Nn+1 = Nn ∪ {n} .
Def

52*D#,B$#2`>H9>H@' H6#2BGA>6'A f1EBZCA/U@ 9!>H9$ CUW (. |A| = n kde n ∈ N, ak |A| = |Nn|.

Lema 1 ∀ n∈N, |Nn| < |Nn+1| .
Veta 1

1!/BGA>6'A fBZCAGU@ 9!>H9MA ∈_/I) "!C+,#2!>HA#Y"AZFJ

Veta 2 Pre n, m ∈ N je |Nn| = |Nm| ⇔ n=m .
Lema 2 Ak A⊆Nn tak ∃ k∈

_g) "!06# |A| = k .

Lema 3

1S!=BGA>H6'A C1 ⊆_h+,#2<&`>H@ CA#>H`@. "A'Y&# ná, tak potom |A| = |N| .

Veta 3

1!/BGA>6'A f1i+,#2!>A#Y"AZ) "! ∃) "!02U@'@>DH<#,A0SY&W (O >=A ∈_j6# |A| = n.

Veta

_$#N"`/BGA>H6'A;=1=5E( -G!>A#Y"A0S CA#N"` |A| = n , |B| = m 2<Z"@>9# 1 ∩B=∅ , potom platí

a)

|A| + |B| = |A∪B| = n + m,

b)

|A| * |B| = |AxB| = n * m,

c)

|A|

|

B| = | AB| = nm.

Lema 4 Nech f je zobrazenie z NxN

D>S_g) "!06# ∀ n, m ∈ N, platí

(1)

f(n,0) = n , g(n,m + 1) = g(n,m) + n , potom

(2)

f(n,m + 1) = f(n,m) + 1 , potom platí

(3)

f(n,m) = n + m .

(4)

n + 0 = n ,

(5)

n + 1 = n + m + 1.

Poznámka Vlastnosti (4) a (5)

+,#DA><&A Y"A#SN"` "@ "!) #,@'<&*,+.-(.Y&') >H9 "A'#2U@'@>DH<#,AXN"`Y&W (#&O J

Lema 5 Nech g je zobrazenie z NxN

DH>S_g) "!0H6# ∀ n, m ∈ N platí

(9)

g(n,0) = 0,

(10) g(n,m + 1) = g(n,m) + n , potom

(11) g(n,m) = n * m.

Lema 6 Nech h: NxN

→ N , ∀ n, m ∈ N platí

(12) h(n,0) = 1

(13) h(n,m + 1) = h(n,m) * n , potom
(14) h(n,m) = n

m.

k #,) 2lS_$#N"`/1/5E( -G!>HA#Y"A0CBGA>'6'A;

|A| = n , |B| = m

a)

Nech A, B sú disjunktné, teda A

∩B = ∅, potom |A∪B| = n + m , <--

U@. "9'DHO >$( -Y,)*

b)

|AxB| = n * m ,

c)

|Am| = nm.

m/n (O#DH>H!$oF1S!/1/5E( -G!>A#Y"A02B/A>6'A;U>)>HBc1 ∪

B, AxB, A

B, P(A)

( -G!>A#Y"A02B/A>6'A;J

Veta 5 Nech Ak, 1

L kLAp(-/BGA>6'A; ) "!0H6#

a)

|Ak| = m pre 1

L kL m

b)

Ak

∩Ak´ = ∅ pre k≠k´, potom |∪k=1 do n, Ak| = m * n .

Veta 6 Nach Ak, 1

L kLAp(-/BGA>6'A; po dvoch disjunktné a |Ak| q n , potom ∀ k∈ (1,n) ⇒ |∪k=1 do n, Ak| q m * n .

Def (m

#DH#,!O ' DH>H9 DH#r 'AW N&' C!>A#Y"A>( ) 'BGA>6WA - Y ( +,#2B$#,A:' jako celok) Hovoríme, 6#2BGA>6'A C1i+,#2!>HA#Y ná ak pre

k

,6D-C+,# +fU>DB/A>6'A*$[ ⊂1/) "!H-H6#][ ≠A platí |X| < |A| .

Veta 7