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III Struktury algebraiczne
KrDZEk
III Struktury algebraiczne
1) Sprawdzi, czy dziaanie
jest dziaaniem wewntrzym w zbiorze A, gdy:
a) a b = ab + a + b,
A = (-1, +),
b) a b = ab - a - b + 2,
A = R \ {1},
c) a b = ab + 2a + 2b + 2,
A = (-2, +),
d) a b = ab + 4a + 4b + 12,
A = R \ {-4},
e) a b = ab + 5a + 5b + 20,
A = (-5, +),
f ) a b = ab - 2a - 2b + 6,
A = R \ {2},
g) a b = ab - 3a - 3b + 12,
A = (3, +).
2) Sprawdzi wasnoci dziaania wewntrznego w zbiorze A i na tej podstawie okrli czy jest
to grupa czy grupa abelowa.
a) A = R,
a b = a + ab + b,
b) A = R,
a b = a + b + 3,
c) A = R,
a b = ab - a - b + 1,
d) A = R,
a b = ab + 5a + 5b + 20,
e) A = (-5, +) ,
a b = ab + 5a + 5b + 20,
f ) A = R,
a b = ab - 3a - 3b + 12,
g) A = (3, +) ,
a b = ab - 3a - 3b + 12,
h) A = R \ {2},
a b = ab - 2a - 2b + 6,
i) A = R \ {1},
a b = ab - a - b + 2,
j) A = R \ {4},
a b = ab - 4a - 4b + 20,
k) A = (-1, +) ,
a b = ab + a + b,
l) A =
2
R ,
x y = (x1, x2) (y1, y2) = (x1 + y1 + 1, x2 + y2),
m) A =
2
R ,
x y = (x1, x2) (y1, y2) = (x1 + y1, x2 + y2 + 2),
n) A =
2
R ,
x y = (x1, x2) (y1, y2) = (x1 + y1 - 2, x2 + y2),
o) A =
2
R ,
x y = (x1, x2) (y1, y2) = (x1 + y1, x2 + y2 - 1),
p) A =
3
R ,
x y = (x1, x2, x3) (y1, y2, y3) = (x1 + y1, x2 + y2, x3 + y3),
r) A =
2
R ,
x y = (x1y2 + x2y1, x2y2),
s) A =
2
R \ {(0, 0)} ,
x y = (x1y1 - x2y2, x1y2 - x2y1).
3) Sprawdzi czy dziaanie
jest rozdzielne wzgldem , jeli:
a) a
b = a + b + 1, a b = a + b + ab,
a, b A = R,
b) x
y = (x
2
1y1, x1y2 + x2y1 + x2y2), x y = (x1 + y1, x2 + y2 + 1),
x, y A = R ,
c) (a, b)
(c, d) = (ac, ad + bc + bd), (a, b) (c, d) = (a + c, b + d),
w
2
R .
Pawe Prysak
Strona 5

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III Struktury algebraiczne
KrDZEk
4) Czy (V, , ) jest przestrzeni wektorow nad ciaem (R, +, *)
V = ( 2
R , R, , )
a) V = R
x y = (x + y + 2),
x = x + 2 - 2,
b) V =
2
R
x y = (x1 + y1, x2 + y2),
x = (x1, x2),
c) V =
2
R
x y = (x1 + y1 + 2, x2 + y2),
x = (x1 + 2 - 2, x2),
d) V =
2
R
x y = (x1 + y1, x2 + y2 - 1),
x = (x1, x2 - + 1, ),
e) V =
2
R
x y = (x1 + y1, x2 + y2 + 2),
x = (x1, x2 + 2 - 2, ),
f ) V =
2
R
x y = (x1 + y1 - 1, x2 + y2),
x = (x1 - + 1, x2).
Pawe Prysak
Strona 6

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