Question

Va rog cele doua exercitii ,dau COROANA.​

Answer 1

1.

A = {x:  x = 2ⁿ  și  n ≤ 2, n ∈ ℕ}

B = {y ∈ ℤ:  y = x-2  și  x ∈ A}

⇒ A = {2⁰, 2¹, 2²} ⇔ A = {1, 2, 4}

⇒ B = {1-2, 2-2, 4-2} ⇔ A = {-1, 0, 2}

a)  A ∪ B = {1, 2, 4} ∪ {-1, 0, 2} = {-1, 0, 1, 2, 4}

b)  A ∩ B = {1, 2, 4} ∩ {-1, 0, 2} = {2}

c)  B - A = {-1, 0, 2} - {1, 2, 4} = {-1, 0}

d)  A × B = {1, 2, 4} × {-1, 0, 2} =

= { (1,-1), (1,0), (1,2), (2,-1), (2,0), (2,2), (4,-1), (4,0), (4,2) }

e)  B × A = {-1, 0, 2} × {1, 2, 4} =

= { (-1,1), (-1,2), (-1,4), (0,1), (0,2), (0,4), (2,1), (2,2), (2,4) }

2.

a)  card(A ∪ B) = 20,  card(A - B) = 8,  card(B - A) = 7

card(A ∪ B) = card(A) + card(B) - card(A ∩ B)

card(A - B) = card(A) - card(A ∩ B)

⇒ card(B)  = card(A ∪ B) - card(A - B)  = 20 - 8 = 12

⇒ card(A) = card(A ∪ B) - card(B - A) = 20 - 7 = 13

b)  card(A ∩ B) = 10,  card(A - B) = 5,  card(B - A) = 11

card(A - B) = card(A) - card(A ∩ B)

⇒ 5 = card(A) - 10 ⇒ card(A) = 15

card(B - A) = card(B) - card(A ∩ B)

⇒ 11 = card(B) - 10 ⇒ card(B) = 21

c)  card(A ∩ B) = 0,  card(A ∪ B) = 28,  card(A - B) = 17

card(A - B) = card(A) - card(A ∩ B)

⇒ 17 = card(A) - 0 ⇒ card(A) = 17

card(A ∪ B) = card(A) + card(B) - card(A ∩ B)

⇒ 28 = 17 + card(B) - 0 ⇒ card(B) = 11