"a" y "b" €R, Si "a" esta en el intervalo [1,2] y "b" en el intervalo [3,4[. ¿En que intervalo se encuentra a/B?
Respuestas
Respuesta:
A = [-3 , 4[
B = ]-∞ , 2[
C = [-1 , 3]
D = [0,+∞[ y
U = R = ]-∞ , +∞[
complemento de un conjunto A
A' = U - A
a) (A ∪ B) - C'
A ∪ B = ]-∞ , 4[
C' = U - C = R - C = R - [-1 , 3]
(A ∪ B) - C' = ]- 1 , 3[
b) (B ∩ C)' ∪ (D - A)
B ∩ C = Ф
(B ∩ C)' = R
D - A = ] 4 , +∞[
(B ∩ C)' ∪ (D - A) = R
c) (A ∪ B) ∪ (C ∩ D) - (D' ∩ B)
A ∪ B = ]-∞ , 4[
C ∩ D = ]0 , 3[
(A ∪ B) ∪ (C ∩ D) = R
D' = R - D = ]-∞ , 0[
D' ∩ B = ]-∞ , 2[
(A ∪ B) ∪ (C ∩ D) - (D' ∩ B) = R - ]-∞ , 2[
d) (A ∪ (B - C)')'
B - C = ]-∞ , 2[
(B - C)' = R - ]-∞ , 2[
A ∪ (B - C)' = [-3 , +∞[
(A ∪ (B - C)')' = R - [-3 , +∞[
e) (B ∪ D) - (A - C)
B ∪ D = ]-∞ , 2[ ∪ ]0 , +∞[
A - C = [-3 , 0[
(B ∪ D) - (A - C) = ]-∞ , -3[ ∪ ]0 , +∞[
f) A' ∩ D'
Aplica Ley de De Morgan
(A ∪ B)' = A' ∩ B'
A U B = ]-∞ , 4[
(A ∪ B)' = U - (A ∪ B) = R - (A ∪ B)
A' ∩ D' = R - ]-∞ , 4[
g) (A ∪ B) - D
h) C ∩ D'
i) (B' ∩ D) ∪ A
B ' = R - ]-∞ , 2[
B' ∩ D = ]-∞ , 2[
(B' ∩ D) ∪ A = [-3 , 4[ -{2} = [-3 , 2[ ∪ ]2 , 4[
j) (A - B) ∪ (U ∩ D)
A - B = ] -2 , -4[
U ∩ D = D = [0 , +∞[
(A - B) ∪ (U ∩ D) = ] -2 , -4[ ∪ [0 , +∞[
k) (A - C)' - D
A - C = [-3 , -1[ ∪ ]3 , 4[
(A - C)' = R - (A - C)' = ]-∞ , -3[ ∪ ]-1 , 3[ ∪ ]4 , +∞[
(A - C)' - D = ]-∞ , -3[ ∪ ]-1 , 0[
l) (A' ∩ D') - B
A U D = [- 3 , +∞[
Ley de De Morgan (A' ∩ D') = (A ∪ D)'
(A ∪ D)' = R - [- 3 , +∞[ = ]-∞ , -3[
(A' ∩ D') - B = ]-3 , -2[
m) ((C' ∩ B') - A)'
C' ∩ B' = (C ∪ B)'
C ∪ B = ]-∞ , -2[ ∪ [-1 , 3]
(C ∪ B)' = R - (]-∞ , -2[ ∪ [-1 , 3])
(C ∪ B)' = ]-2 , -1[ ∪ [3 , +∞]
(C' ∩ B') -