ÿþR o z d z i a B 2
R a c h u n e k z b i o r ó w
E l |b i e t a K o t l i c k a , J o a n n a R z e p e c k a
2 . 1 . W s t a w s y m b o l " l u b ‚" o r a z o d p o w i e d n i e o k r e [l e n i e t a k , a b y u z y s k a z d a n i e p r a w d z i w e :
a ) - 2 " Z , t z n . - 2 j e s t . . . . . . . . . . . . . . . . . . z b i o r u l i c z b c a l k o w i t y c h ;
b ) 1 . . . [ 1 , 2 ) , t z n . 1 n a l e |y d o p r z e d z i a l u [ 1 , 2 ) ;
c ) N . . . R , t z n . z b i ó r l i c z b . . . . . . . . . . . . . . . . . . j e s t p o d z b i o r e m z b i o r u l i c z b r z e c z y w i s t y c h ;
3 3
d ) . . . Q , t z n . j e s t . . . . . . . . . . . . . . . . . . z b i o r u l i c z b w y m i e r n y c h ;
2 2
e ) " . . . { 1 , 2 , 3 } , t z n . z b i ó r . . . . . . . . . . . . . . . j e s t . . . . . . . . . . . . . . . . . . z b i o r u { 1 , 2 , 3 } ;
f ) a . . . { a , b , c } , t z n . a j e s t . . . . . . . . . . . . . . . . . . z b i o r u { a , b , c } ;
g ) { a } . . . { a , b , c } , t z n . { a } j e s t . . . . . . . . . . . . . . . . . . z b i o r u { a , b , c } .
2 . 2 . W s t a w s y m b o l " l u b ‚" t a k , a b y u z y s k a z d a n i e p r a w d z i w e :
a ) 1 . . . Q ; f ) { "} . . . { ", 1 , - 1 } ;
b ) { 1 } . . . Z ; g ) " . . . { ", 1 , - 1 } ;
c ) { - 1 , 0 , 1 } . . . { - 1 , 0 , 1 } ; h ) { a , b } . . . { { a } , { a , b } , { a , b , c } } ;
d ) N . . . Z ; i ) { 0 , 3 } . . . [ 0 , 3 ] ;
e ) N . . . { N , Z , R } ; j ) ( 0 , 1 ) . . . [ 0 , 2 ] .
2 . 3 . W s t a w s y m b o l " l u b " t a k , a b y u z y s k a z d a n i e p r a w d z i w e :
/
"
a ) 0 . . . N ; f ) 4 . . . Q ;
6
b ) 2 - 1 . . . N ; g ) . . . Z ;
3
3
c ) À . . . Q ; h ) . . . Z ;
6
d ) s i n À . . . Q ; i ) 0 . . . [ 0 , 2 ) ;
"
e ) 2 . . . Q ; j ) 2 . . . [ 0 , 2 ) .
2 E l |b i e t a K o t l i c k a , J o a n n a R z e p e c k a
2 . 4 . U z u p e l n i j p o n i |s z e z d a n i a , u |y w a j a c s y m b o l u = ( g d y p o d a n e z b i o r y s a Û r ó w n e ) o r a z s y m b o l i ‚" i ƒ"
Û
w p r z e c i w n y m w y p a d k u :
a ) ( - ", 1 ) . . . { x " R : x d" 1 } ; e ) ( 0 , 1 ) . . . { x " R : x e" 0 '" x < 1 } ;
b ) [ 2 , + ") . . . { x " R : x e" 2 } ; f ) Z . . . { x " R : | x | = 5 } ;
c ) ( 3 , ") . . . { x " R : <" ( x < 3 ) } ; g ) { x " R : x 2 e" 1 } . . . { x " R : | x | e" 1 } ;
d ) [ 0 , 1 ] . . . { x " R : x e" 0 '" x d" 1 } ; h ) { x " R : x 2 e" 2 } . . . { x " R : | x | e" 2 } .
2 . 5 . W y z n a c z w s z y s t k i e p o d z b i o r y z b i o r u X :
a ) X = "; c ) X = { a , b , c } ; e ) X = { ", a } ;
b ) X = { 1 } ; d ) X = { 1 , 2 , 3 , 4 } ; f ) X = { { 1 } , { 1 , 2 } } .
J a k a j e s t z a l e |n o [ m i d z y l i c z b a Û e l e m e n t ó w d a n e g o z b i o r u a i l o [c i a Û j e g o p o d z b i o r ó w ?
2 . 6 . W y z n a c z p o d a n a Û s u m z b i o r ó w :
a ) { 0 , 1 } *" { 2 , 3 } ; d ) ( 0 , 2 ) *" ( 1 , + ") ; g ) R *" [ 0 , 1 ] ;
b ) { 0 , 1 } *" { 0 , 1 , 2 } ; e ) ( 0 , 2 ) *" [ 2 , 3 ] ; h ) " *" [ 0 , 1 ] ;
c ) N *" { 1 , 2 , 3 } ; f ) ( 0 , 2 ) *" [ 0 , 3 ] ; i ) [ 0 , 1 ] *" [ 0 , 1 ] .
2 . 7 . W y z n a c z p o d a n y i l o c z y n ( c z [ w s p ó l n a ) z b i o r ó w :
Û
a ) { 0 , 1 } )" { 0 , 1 , 2 } ; d ) ( 0 , 1 ) )" [ 1 , 3 ] ; g ) " )" [ 0 , 1 ] ;
b ) N )" { 0 , 1 , 2 , 3 } ; e ) ( 0 , 1 ] )" [ 1 , 3 ] ; h ) [ 0 , 1 ] )" [ 0 , 1 ] ;
c ) ( 0 , 2 ) )" ( 1 , 3 ) ; f ) R )" [ 0 , 1 ] ; i ) ( - 1 , + ") )" [ 2 , + ") .
2 . 8 . W y z n a c z p o d a n a Û r ó |n i c z b i o r ó w :
a ) { 0 , 1 } \ { 2 , 3 } ; d ) ( 0 , 2 ) \ ( 1 , + ") ; g ) [ 0 , 1 ] \ ";
b ) { 0 , 1 } \ { 0 , 1 , 2 } ; e ) ( 0 , 1 ) \ [ 1 , 3 ] ; h ) " \ [ 0 , 1 ] ;
c ) N \ { 0 , 1 , 2 , 3 } ; f ) [ 1 , 3 ] \ ( 1 , 2 ) ; i ) [ 0 , 1 ] \ [ 0 , 1 ] .
2 . 9 . D l a d a n e g o z b i o r u A w y z n a c z j e g o d o p e l n i e n i e d o z b i o r u R , t z n . z b i ó r A = R \ A :
a ) A = ( - ", 1 ) ; d ) A = "; g ) A = ( - ", 2 ) *" ( 2 , + ") ;
b ) A = ( 0 , 2 ) ; e ) A = R ; h ) A = { 0 , 1 } ;
c ) A = [ 0 , 2 ] ; f ) A = ( - 1 , 0 ] *" { 3 } ; i ) A = N .
2 . 1 0 . W y z n a c z z b i o r y : A *" B , A )" B , A \ B o r a z B \ A , j e [l i :
a ) A = [ 0 , 2 ] , B = ( 0 , 2 ) ; d ) A = [ - 2 , 1 ] , B = [ 1 , 5 ] ;
b ) A = ( 0 , + ") , B = [ 0 , 1 ] ; e ) A = ( - ", - 1 ] *"( 3 , + ") , B = { - 1 , 0 , 3 } ;
c ) A = ( - ", 1 ] , B = ( 1 , + ") ; f ) A = ( - ", 1 ] *"[ 3 , 5 ) , B = ( 0 , 4 ] *"[ 6 , + ") .
2 . R a c h u n e k z b i o r ó w 3
2 . 1 1 . W y z n a c z p o d a n e z b i o r y , j e [l i A = ( 1 , 3 ) , B = [ 2 , 4 ] i C = ( 0 , + ") :
a ) ( A )" B ) , ( A *" B ) , A )" B , A *" B ;
b ) A )" ( B *" C ) , ( A )" B ) *" C , ( A *" B ) )" C , ( A )" B ) *" ( A )" C ) .
K t ó r e z w y z n a c z o n y c h z b i o r ó w s a Û r ó w n e ?
2 . 1 2 . U d o w o d n i j , |e d l a d o w o l n y c h z b i o r ó w A , B , C ‚" X z a c h o d z a Û r ó w n o [c i :
a ) ( A )" B ) = A *" B ; d ) A )" ( B *" C ) = ( A )" B ) *" ( A )" C ) ;
b ) ( A *" B ) = A )" B ; e ) A *" ( B )" C ) = ( A *" B ) )" ( A *" C ) ;
c ) ( A \ B ) = A *" B ; f ) A \ ( B *" C ) = ( A \ B ) \ C .
2 . 1 3 . C z y d l a d o w o l n y c h z b i o r ó w A , B , C ‚" X z a c h o d z a Û p o n i |s z e r ó w n o [c i ? U z a s a d n i j o d p o w i e d z.
a ) A \ ( B )" C ) = ( A \ B ) )" ( A \ C ) ; d ) ( A *" B ) \ B = A ;
b ) A *" ( B \ C ) = ( A *" B ) \ ( A )" C ) ; e ) ( A \ B ) \ C = A \ ( B \ C ) ;
c ) A \ B = ( A *" B ) ; f ) A )" ( B \ C ) = ( A )" B ) \ ( A )" C ) .
2 . 1 4 . W y z n a c z z b i o r y : A *" B , A )" B , A \ B o r a z B \ A , j e [l i :
a ) A = [ 0 , 2 ] , B = { x " R : | x - 1 | e" 1 } ;
b ) A = ( 0 , + ") , B = { x " R : x ( x - 1 ) d" 0 } ;
c ) A = N , B = { x " R : | x - 2 | d" 3 } ;
d ) A = { x " R : | x + 1 | e" 1 } , B = { x " R : | x + 1 | d" 3 } ;
e ) A = { x " R : x 2 + 2 x - 3 = 0 } , B = { x " R : x 2 + 2 x - 3 e" 0 } ;
f ) A = { x " R : x 2 + 2 x + 1 d" 0 } , B = { x " R : x 2 + 2 x + 1 > 0 } .
2 . 1 5 . W y z n a c z p o d a n e z b i o r y :
A = { x " R : | x | e" 1 (" | x | > 1 } ;
B = { x " R : | x | > 1 '" | x | > 3 } ;
C = { x " R : x + 2 e" 0 (" x 2 > - 1 (" 4 - x < 3 } ;
D = { x " R : x 2 > 1 (" x 3 > 1 (" x 4 > 1 } ;
E = { x " R : x > 1 '" | x | > 3 } ;
F = { x " Z : 1 < | x | < 5 } ;
G = { x " N : 1 < | x | < 5 } ;
H = { x " Q : x 2 + 4 x = 4 } ;
I = { x " R : x 2 + 4 x = 5 '" x e" 0 } ;
J = { x " R : x > 1 '" | 2 x | d" 4 '" x 2 > 5 } ;
K = { x " R : ( x > 1 '" | 2 x | d" 4 ) (" x 2 > 5 } .
2 . 1 6 . W y z n a c z ( n a r y s u j ) z b i o r y : A × B , B × A , A 2 = A × A o r a z B 2 = B × B , j e [l i :
a ) A = { - 2 , 3 } , B = [ 1 , 4 ) ;
b ) A = { - 2 , 2 , 4 } , B = ( 1 , + ") ;
c ) A = [ - 1 , 2 ] , B = ( 1 , 5 ) .
4 E l |b i e t a K o t l i c k a , J o a n n a R z e p e c k a
2 . 1 7 . U d o w o d n i j , |e d l a d o w o l n y c h z b i o r ó w A , B , C ‚" X z a c h o d z a Û r ó w n o [c i :
a ) A × ( B *" C ) = ( A × B ) *" ( A × C ) ;
b ) ( A )" B ) × C = ( A × C ) )" ( B × C ) ;
c ) ( A \ C ) × B = ( A × B ) \ ( C × B ) .
2 . 1 8 . K t ó r y z p r z e d s t a w i o n y c h p o n i |e j z b i o r ó w m o |n a p r z e d s t a w i w p o s t a c i i l o c z y n u k a r t e z j a Ds k i e g o ?
y y
A E
2
1 1
1 2 - 2 1
x x
y
F
y
B
2
1
- 2 2
x
1 2
x
- 2
y
G
y
C
4
2
1
1 3
x
- 3 - 2 1 3
x
y y
D H
3
1 1
1 3 1
x x
2 . R a c h u n e k z b i o r ó w 5
2 . 1 9 . N a s z k i c u j p o d a n e z b i o r y o r a z o p i s z i n a s z k i c u j i c h d o p e l n i e n i a d o p r z e s t r z e n i R 2 .
A = ( x , y ) " R 2 : y = x ; C = ( x , y ) " R 2 : y d" | x | ;
B = ( x , y ) " R 2 : y < x ; D = ( x , y ) " R 2 : x 2 + y 2 e" 4 .
2 . 2 0 . N a s z k i c u j p o d a n e z b i o r y . J e [l i t o m o |l i w e z a p i s z j e w p o s t a c i i l o c z y n u k a r t e z j a Ds k i e g o .
A = { ( x , y ) " R 2 : x y > 0 } ; E = { ( x , y ) " Z 2 : x 2 + y 2 - 2 x + 2 y < 2 } ;
B = { ( x , y ) " R 2 : ( x + 2 ) 2 + ( y - 1 ) 2 = 1 } ; F = { ( x , y ) " R 2 : | x | + | y | d" 3 } ;
C = { ( x , y ) " R 2 : ( x + 2 ) 2 + ( y - 1 ) 2 = 0 } ; G = { ( x , y ) " R 2 : x 2 + x - 2 d" 0 } ;
D = { ( x , y ) " R 2 : x 2 + y 2 - 2 x + 2 y < 2 } ; H = { ( x , y ) " R 2 : | y | > 2 } .
2 . 2 1 . W y z n a c z ( n a s z k i c u j ) z b i o r y : A *" B , A )" B , A \ B o r a z B \ A , j e [l i :
a ) A = [ - 2 , 2 ] × [ 0 , 2 ] , B = R × ( 1 , 4 ] ;
b ) A = ( x , y ) " R 2 : y d" x 3 , B = { ( x , y ) " R 2 : y > x 3 } ;
c ) A = { ( x , y ) " R 2 : y 2 - 2 y d" 0 } , B = { ( x , y ) " R 2 : y 2 - 2 y = 0 } ;
d ) A = { ( x , y ) " R 2 : | x - 1 | e" 2 } , B = { ( x , y ) " R 2 : ( x - 1 ) 2 + y 2 d" 4 } .
2 . 2 2 . W y z n a c z ( n a s z k i c u j ) p o d a n e z b i o r y :
1
A = { ( x , y ) " R 2 : y > x } )" { ( x , y ) " R 2 : y > x 2 } ;
2
B = [ - 2 , 2 ] 2 \ { ( x , y ) " R 2 : x 2 + y 2 e" 4 } ;
C = { ( x , y ) " R 2 : x 2 + y 2 + 2 x + 2 y + 1 d" 0 } *" { ( x , y ) " R 2 : x 2 + y 2 - 2 x - 2 y + 1 d" 0 } *"
*"{ ( x , y ) " R 2 : x 2 + y 2 = 1 } ;
D = { ( x , y ) " R 2 : y > x } )" { ( x , y ) " R 2 : y < 2 x } )" { ( x , y ) " R 2 : | x | d" 2 } ;
E = { ( x , y ) " R 2 : y e" x 2 + 1 } *" { ( x , y ) " R 2 : y d" 1 - | x | } )" { ( x , y ) " R 2 : | y | < 2 } ;
F = { ( x , y ) " R 2 : y d" - x 2 + 2 x } *" { ( x , y ) " R 2 : y d" - x 2 - 2 x } )" { ( x , y ) " R 2 : y e" | x | - 2 } ;
G = { ( x , y ) " R 2 : x 2 + y 2 < 6 x } )" { ( x , y ) " R 2 : x 2 + y 2 > 4 x } *" { ( x , y ) " R 2 : x 2 + y 2 d" 2 x } ;
H = { ( x , y ) " R 2 : x 2 + y 2 d" 4 } \ { ( x , y ) " R 2 : x 2 + y 2 d" 3 } *" { ( x , y ) " R 2 : y = x 2 + 2 } .
2 . 2 3 . W y z n a c z ( n a s z k i c u j ) p o d a n e z b i o r y :
A = { ( x , y ) " R 2 : 2 x 2 d" y d" 3 - x 2 } ;
B = { ( x , y ) " R 2 : y > x 2 '" x > y 2 } ;
C = { ( x , y ) " R 2 : y d" x 2 '" x 2 + y 2 d" 2 y '" x > 0 } ;
D = { ( x , y ) " R 2 : y > | x | (" y d" 1 - | x - 1 | } ;
E = { ( x , y ) " R 2 : x 2 - 2 x + 1 + y 2 = 0 (" x 2 + y 2 e" 2 x } ;
1 1
F = { ( x , y ) " R 2 : y d" x 2 - 2 (" y > 2 - x 2 (" x 2 + y 2 d" 1 } ;
8 8
G = { ( x , y ) " R 2 : ( x 2 d" 2 x '" y < 3 ) (" y = 3 } ;
H = { ( x , y ) " R 2 : ( y > x 2 + 1 (" y d" - x 2 - 1 ) '" x 2 + y 2 < 1 } .
6 E l |b i e t a K o t l i c k a , J o a n n a R z e p e c k a
O d p o w i e d z i
2 . 1 . a ) e l e m e n t e m d ) ", e l e m e n t e m g ) ‚", p o d z b i o r e m
b ) " e ) ‚", p u s t y , p o d z b i o r e m
c ) ‚", n a t u r a l n y c h f ) ", e l e m e n t e m
2 . 2 . a ) " d ) ‚" g ) " j ) ‚"
b ) ‚" e ) " h ) "
c ) ‚" f ) ‚" i ) ‚"
2 . 3 . a ) " d ) " g ) " j ) "
/ /
b ) " e ) " h ) "
/ / /
c ) " f ) " i ) "
/
2 . 4 . a ) ‚" c ) ‚" e ) ‚" g ) =
b ) = d ) = f ) ƒ" h ) ƒ"
2 . 5 . a ) "
b ) ", { 1 }
c ) ", { a } , { b } , { c } , { a , b } , { b , c } , { a , c } , { a , b , c }
d ) ", { 1 } , { 2 } , { 3 } , { 4 } , { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } , { 1 , 2 , 3 } , { 1 , 2 , 4 } , { 1 , 3 , 4 } , { 2 , 3 , 4 } ,
{ 1 , 2 , 3 , 4 }
e ) ", { "} , { a } , { ", a }
f ) ", { { 1 } } , { { 1 , 2 } } , { { 1 } , { 1 , 2 } }
J e [l i z b i ó r X p o s i a d a n e l e m e n t ó w , t o i s t n i e j e 2 n j e g o p o d z b i o r ó w ( z t e g o p o w o d u z b i ó r p o d z b i o r ó w
z b i o r u X n a z y w a n y j e s t z b i o r e m p o t g o w y m i n a j c z [c i e j o z n a c z a n y p r z e z 2 X ) .
2 . 6 . a ) { 0 , 1 , 2 , 3 } d ) ( 0 , + ") g ) R
b ) { 0 , 1 , 2 } e ) ( 0 , 3 ] h ) [ 0 , 1 ]
c ) N f ) [ 0 , 3 ] i ) [ 0 , 1 ]
2 . 7 . a ) { 0 , 1 } d ) " g ) "
b ) { 1 , 2 , 3 } e ) { 1 } h ) [ 0 , 1 ]
c ) ( 1 , 2 ) f ) [ 0 , 1 ] i ) [ 2 , + ")
2 . 8 . a ) { 0 , 1 } d ) ( 0 , 1 ] g ) [ 0 , 1 ]
b ) " e ) ( 0 , 1 ) h ) "
c ) { 4 , 5 , 6 , . . . } f ) { 1 } *" [ 2 , 3 ] i ) "
2 . R a c h u n e k z b i o r ó w 7
2 . 9 . a ) A = [ 1 , + ") f ) A = ( - ", - 1 ] *" ( 0 , 3 ) *" ( 3 , + ")
g ) A = { 2 }
b ) A = ( - ", 0 ] *" [ 2 , + ")
h ) A = ( - ", 0 ) *" ( 0 , 1 ) *" ( 1 , + ")
c ) A = ( - ", 0 ) *" ( 2 , + ")
i ) A = ( - ", 1 ) *" ( 1 , 2 ) *" ( 2 , 3 ) *" . . .
d ) A = R
e ) A = "
2 . 1 0 . a ) A *" B = [ 0 , 2 ] , A )" B = ( 0 , 2 ) , A \ B = { 0 , 2 } , B \ A = "
b ) A *" B = [ 0 , + "] , A )" B = ( 0 , 1 ] , A \ B = ( 1 , + ") , B \ A = { 0 }
c ) A *" B = R , A )" B = ", A \ B = ( - ", 1 ] , B \ A = ( 1 , + ")
d ) A *" B = [ - 2 , 5 ] , A )" B = { 1 } , A \ B = [ - 2 , 1 ) , B \ A = ( 1 , 5 ]
e ) A *"B = ( - ", - 1 ] *" { 0 } *"[ 3 , + ") , A )"B = { - 1 } , A \ B = ( - ", - 1 ) *"( 3 , + ") , B \ A = { 0 , 3 }
f ) A *"B = ( - ", 5 ) *"[ 6 , + ") , A )"B = ( 0 , 1 ] *"[ 3 , 4 ] , A \ B = ( - ", 0 ] *"( 4 , 5 ) , B \ A = ( 1 , 3 ) *"[ 6 , + ")
2 . 1 1 . a ) ( A )" B ) = [ 2 , 3 ) = ( - ", 2 ) *" [ 3 , + ") = A *" B
( A *" B ) = ( 1 , 4 ] = ( - ", 1 ] *" ( 4 , + ") = A )" B
b ) A )" ( B *" C ) = ( 1 , 3 ) )" ( 0 , + ") = ( 1 , 3 ) = ( A )" B ) *" ( A )" C )
( A )" B ) *" C = [ 2 , 3 ) *" ( 0 , + ") = ( 0 , ")
( A *" B ) )" C = ( 1 , 4 ] )" ( 0 , + ") = ( 1 , 4 ]
2 . 1 2 . A b y w y k a z a r ó w n o [ d w ó c h z b i o r ó w E i F n a l e |y u d o w o d n i , |e d l a k a |d e g o x z a c h o d z i r ó w n o w a |n o [:
( x " E Ô! x " F ) . W p o n i |e j p r z e d s t a w i o n y c h d o w o d a c h k o l e j n e r ó w n o w a |n e f o r m u l y p o w s t a j a Û w
w y n i k u z a s t o s o w a n i a d e f i n i c j i d z i a l a D n a z b i o r a c h i p r a w r a c h u n k u z d a D t a k i c h j a k : p r a w a d e M o r g a n a
( P d M ) , p r a w a r o z d z i e l n o [c i ( P R ) , p r a w a l a c z n o [c i ( P L ) o r a z p r a w o p o d w ó j n e g o z a p r z e c z e n i a ( P P Z ) .
Û
a ) x " ( A )" B ) Ô! ( x " X \ ( A )" B ) ) Ô! ( x " X '" <" ( x " A )" B ) ) Ô!
( P d M ) ( P R )
Ô! ( x " X '" <" ( x " A '" x " B ) ) Ô! ( x " X '" ( <" x " A (" <" x " B ) ) Ô!
Ô! ( ( x " X '" <" x " A ) (" ( x " X '" <" x " B ) ) Ô! ( x " X \ A (" x " X \ B ) Ô!
Ô! ( x " A (" x " B ) Ô! x " A *" B
b ) d o w ó d a n a l o g i c z n y j a k w p u n k c i e a )
c ) x " ( A \ B ) Ô! x " X \ ( A \ B ) Ô! ( x " X '" <" ( x " A \ B ) ) Ô!
( P d M ) ( P P Z )
Ô! ( x " X '" <" ( x " A '" <" x " B ) ) Ô! Ô! ( x " X '" ( <" x " A (" <" ( <" x " B ) ) ) Ô!
( P R )
Ô! ( x " X '" ( <" x " A (" x " B ) ) Ô! Ô! ( ( x " X '" <" x " A ) (" ( x " X '" x " B ) ) Ô!
Ô! ( x " X \ A (" x " X )" B ) Ô! ( x " A (" x " B ) Ô! x " A *" B
( P R )
d ) x " A )" ( B *" C ) Ô! ( x " A '" x " B *" C ) Ô! ( x " A '" ( x " B (" x " C ) ) Ô!
Ô! ( ( x " A '" x " B ) (" ( x " A '" x " C ) ) Ô! ( x " A )" B (" x " A )" C ) Ô!
Ô! x " ( A )" B ) *" ( A )" C )
e ) d o w ó d a n a l o g i c z n y j a k w p u n k c i e d )
( P d M )
f ) x " A \ ( B *" C ) Ô! ( x " A '" <" x " B *" C ) Ô! ( x " A '" <" ( x " B (" x " C ) ) Ô!
( P L )
Ô! ( x " A '" ( <" x " B '" <" x " C ) ) Ô! Ô! ( ( x " A '" <" x " B ) '" <" x " C ) ) Ô!
Ô! ( x " A \ B ) '" <" x " C ) Ô! x " ( A \ B ) \ C
8 E l |b i e t a K o t l i c k a , J o a n n a R z e p e c k a
2 . 1 3 . M o |n a u d o w o d n i ( p a t r z p o p r z e d n i e z a d a n i e ) , |e z a c h o d z a Û r ó w n o [c i w p u n k t a c h c ) i f ) .
W p o z o s t a l y c h p r z y p a d k a c h r o z w a |a n e r ó w n o [c i n i e z a w s z e s a Û p r a w d z i w e , n p . r ó w n o [ a ) n i e z a c h o d z i
d l a z b i o r ó w : A = { 0 , 1 , 2 } , B = { 1 , 2 } i C = { { 1 , 3 } , g d y | A \ ( B )" C ) = { 0 , 1 , 2 } \ ( { 1 , 2 } )" { 1 , 3 } ) =
{ 0 , 1 , 2 } \ { 1 } = { 0 , 2 } , z a [ ( A \ B ) )"( A \ C ) = ( { 0 , 1 , 2 } \ { 1 , 2 } ) )"( { 0 , 1 , 2 } \ { 1 , 3 } ) = { 0 } )"{ 0 , 2 } = { 0 } .
2 . 1 4 . a ) A *" B = R , A )" B = { 0 , 2 } , A \ B = ( 0 , 2 ) , B \ A = ( - ", 0 ) *" ( 2 , + ")
b ) A *" B = [ 0 , + ") , A )" B = ( 0 , 1 ] , A \ B = ( 1 , + ") , B \ A = { 0 }
c ) A *" B = [ - 1 , 5 ] *" { 6 , 7 , 8 , . . . } , A )" B = { 1 , 2 , 3 , 4 , 5 } , A \ B = { 6 , 7 , 8 , . . . } ,
B \ A = ( - 1 , 1 ) *" ( 1 , 2 ) *" ( 2 , 3 ) *" ( 3 , 4 ) *" ( 4 , 5 )
d ) A *" B = R , A )" B = [ - 4 , - 2 ] *" [ 0 , 2 ] , A \ B = ( - ", - 4 ) *" ( 2 , + ") , B \ A = ( - 2 , 0 )
e ) A *" B = ( - ", - 3 ] *" [ 1 , + ") , A )" B = { - 3 , 1 } , A \ B = ", B \ A = ( - ", - 3 ) *" ( 1 , + ")
f ) A *" B = R , A )" B = ", A \ B = { - 1 } , B \ A = ( - ", - 1 ) *" ( - 1 , + ")
2 . 1 5 . a ) ( - ", - 1 ] *" [ 1 , + ") g ) { 2 , 3 , 4 }
b ) ( - ", - 3 ) *" ( 3 , + ")
h ) "
c ) R
i ) { 1 }
d ) ( 1 , + ")
j ) "
e ) ( 4 , + ")
" "
f ) { - 4 , - 3 , - 2 , 2 , 3 , 4 } k ) - ", - 5 *" ( 1 , 2 ] *" 5 , + "
2 . 1 6 . A × B B × A A × A B × B
y y y y
a )
4 4
3 3
1 1 1
- 2 3 1 4 - 2 3 1 4
x x x x
- 2 - 2
y y y y
b )
4 4
2 2
1 1
- 2 2 4 1 - 2 2 4 1
x x x x
- 2 - 2
y y y y
c )
5 5
2 2
1 1
- 1 2 1 5 - 1 2 1 5
x - 1 x x x
2 . 1 7 . D o w o d y i o z n a c z e n i a p o d o b n e j a k w z a d a n i u 2 . 1 2 . D o d a t k o w o w y k o r z y s t u j e m y p r a w a p r z e m i e n n o [c i
( P P ) o r a z n a s t p u j a c e t a u t o l o g i e : [ ( p '"q ) Ô! ( p '"q '"q ) ] ( P 1 ) , [ ( p '"F ) Ô! F ] ( P 2 ) o r a z [ ( p ("F ) Ô! p ]
Û
( P 3 ) ( w ( P 1 ) i ( P 2 ) F o z n a c z a f o r m u l z a w s z e f a l s z y w a ) .
Û
2 . R a c h u n e k z b i o r ó w 9
( P R )
a ) ( x , y ) " A × ( B *" C ) Ô! ( x " A '" y " B *" C ) Ô! ( x " A '" ( y " B (" y " C ) ) Ô!
Ô! ( ( x " A '" y " B ) (" ( x " A '" y " C ) ) Ô! Ô! ( x " A × B (" x " A × C ) Ô!
Ô! x " ( A × B ) *" ( A × C )
( P 1 )
b ) ( x , y ) " ( A )" B ) × C Ô! ( x " A )" B '" y " C ) Ô! ( ( x " A '" x " B ) '" y " C ) Ô!
( P L ) ( P P )
Ô! ( ( x " A '" x " B ) '" y " C '" y " C ) Ô! ( x " A '" x " B '" y " C '" y " C ) Ô!
( P L )
Ô! ( x " A '" y " C '" x " B '" y " C ) Ô! ( ( x " A '" y " C ) '" ( x " B '" y " C ) ) Ô!
Ô! ( ( x , y ) " A × C (" ( x , y ) " B × C ) Ô! ( x , y ) " ( A × C ) *" ( B × C )
c ) ( x , y ) " ( A × C ) \ ( B × C ) Ô! ( ( x , y ) " A × C ) '" <" ( x , y ) " B × C ) Ô!
( P d M )
Ô! ( ( x " A '" y " C ) '" <" ( x " B '" y " C ) ) Ô!
( P R , P L )
Ô! ( ( x " A '" y " C ) '" ( <" x " B (" <" y " C ) ) Ô!
( P P )
Ô! ( ( x " A '" y " C '" <" x " B ) (" ( x " A '" y " C '" <" y " C ) ) Ô!
( P 2 ) ( P 3 )
Ô! ( ( x " A '" <" x " B '" y " C ) (" ( x " A '" F ) ) Ô! ( ( x " A \ B '" y " C ) (" F ) Ô!
Ô! ( x " A \ B '" y " C ) Ô! ( x , y ) " ( A \ B ) × C
2 . 1 8 . B = { 2 } × R , C = ( [ - 3 , - 2 ] *" ( 1 , 3 ) ) × { 2 } , E = [ - 2 , 1 ] × R , H = R × ( 1 , + ") .
P o z o s t a l y c h z b i o r ó w n i e d a s i p r z e d s t a w i w p o s t a c i i l o c z y n u k a r t e z j a Ds k i e g o .
y y
2 . 1 9 . A A = R 2 \ A =
= { ( x , y ) " R 2 : <" ( y = x ) } =
1 1
= { ( x , y ) " R 2 : y = x }
1 1
x x
y y
B B = { ( x , y ) " R 2 : y e" x }
1
1
1
1 x
x
y
y
C C = { ( x , y ) " R 2 : y > | x | }
1
1
1
1
x x
y y
D D = { ( x , y ) " R 2 : x 2 + y 2 < 4 }
2 2
- 2 2 - 2 2
x x
- 2 - 2
1 0 E l |b i e t a K o t l i c k a , J o a n n a R z e p e c k a
y y
2 . 2 0 . A E
1
1
1
1 2
x
x
- 1
- 2
y y
B F
3
1
- 2 1 - 3 3
x x
- 3
y y
C G
1
1
- 2 1
- 2 1
x
x
y y
D H
2
1
1 1
x x
- 1
- 2
2 . 2 1 . w y p e l n i e n i e z b i o r u A , w y p e l n i e n i e z b i o r u B
A *" B A )" B A \ B B \ A
y y y y
a )
4 4 4 4
3 3 3 3
1 1 1 1
- 2 2 - 2 2 - 2 2 - 2 2
x x x x
b ) y y
R 2 "
1 1
1 1
x x
2 . R a c h u n e k z b i o r ó w 1 1
A *" B A )" B A \ B B \ A
c ) y y y
2 2 2
"
1 1 1
x x x
y y y y
d )
2 2 2 2
- 1 1 3 x - 1 1 3 x - 1 1 3 x - 1 1 3 x
y y
2 . 2 2 . A E
2
2
1
- 3 - 1 1 3
x
2
x
- 2
y y
B F
2
1
- 2 - 1 1 2
x
- 2 2
x
- 2
- 2
y y
C G
3
2 2
1
1
1 2 3 4 6
- 1 x
- 2 - 1 1 2
x
- 2
- 2
- 3
y y
D H
4
2
2
- 2 2
x
2
x
- 2
1 2 E l |b i e t a K o t l i c k a , J o a n n a R z e p e c k a
y y
2 . 2 3 . A E
3
1
2
1 2
x
- 1
- 1 1
x
y
B
y
F
2
1
1
- 4 - 1 1 4
x
1
x
- 2
y
C
y
G
2
3
1
1
- 1 1 2
x
- 1 1
x
y H = "
D
1
1 2
x
Wyszukiwarka
Podobne podstrony:
rachunek zbiorow 2rachunek zbiorow 1rachunek zbiorow 6rachunek zbiorow 7rachunek zbiorow 304 Rachunek zbiorówmat pom Rachunek zbiorowrachunek zbiorow 501 Podstawowe pojecia rachunku zbiorowrachunek zbiorow 4Zasady rachunkowości w zakresie prawa podatkowego w Polscewięcej podobnych podstron