Question

Va rog ex. 15 si 16!!!

Answer 1

Răspuns:

a). Xmin=2¹°°-1, Ymax=3¹°°-1

B) cardA=2¹°²-2¹°°, cardB=3¹°°-3 la98

C) cardB>cardA

Answer 2

Explicație pas cu pas:

15.a)

$A = \Big\{x \Big| x \in \mathbb{N} \ , \ {2}^{100} < x \leqslant {2}^{102} \Big\}$

$\implies {2}^{100} + 1 \leqslant x \leqslant {2}^{102} \Big\}$

→ cel mai mic element al mulțimii A este:

2¹⁰⁰ + 1

$B = \Big\{x \Big| x \in \mathbb{N} \ , \ {3}^{98} \leqslant x < {3}^{100} \Big\}$

$\implies \ {3}^{98} \leqslant x \leqslant {3}^{100} - 1$

→ cel mai mare element al mulțimii B este:

3¹⁰⁰ - 1

b)

card(A) = 2¹⁰² - (2¹⁰⁰ + 1) + 1 = 2¹⁰² - 2¹⁰⁰ = 2¹⁰⁰×(2² - 1) = 3×2¹⁰⁰

card(B) = 3¹⁰⁰ - 1 - 3⁹⁸ + 1 = 3¹⁰⁰ - 3⁹⁸ = 3⁹⁸×(3² - 1) = 8×3⁹⁸

c)

card(A) = 3×2¹⁰⁰ = 3×2³×2⁹⁷ = 24×2⁹⁷

card(B) = 8×3⁹⁸ = 3×8×3⁹⁷ = 24×3⁹⁷

24×2⁹⁷ < 24×3⁹⁷ => card(A) < card(B)

=> mulțimea B are mai multe elemente