Question

A = 4√n unde napartine lui N*, n este mai mic sau = 2022. Determinați A intersectat cu N, A intersectat cu Q, cand A/Q.​

Answer 1

Explicație pas cu pas:

$1 \leqslant n \leqslant 2022 \iff 1 \leqslant \sqrt{n} \leqslant \sqrt{2022}$

$44 < \sqrt{2022} < 45$

$A = \Big\{ 4; 4 \sqrt{2}; 4 \sqrt{3}; 8;... ; 4 \sqrt{1935} ;176; 4 \sqrt{1937}; ...; 4 \sqrt{2021} ; 4 \sqrt{2022} \Big\}$

$\sqrt{n} \in \mathbb{N} \iff \sqrt{n} \in \mathbb{N} \implies \\ n \in \Big\{ 1; {2}^{2}; {3}^{2}; {4}^{2}; ...; {43}^{2}; {44}^{2} \Big\}$

$A \cap \mathbb{N} = \Big\{ 4; 8; 12; 16; ...; 174; 176\Big\}$

$A \cap \mathbb{Q} = A \cap \mathbb{N}$

$card(A) = 2022$

$card(A \cap \mathbb{Q}) = card(A \cap \mathbb{N}) = 44$

$card(A - \mathbb{Q}) = card(A) - card(A \cap \mathbb{Q}) = \\ = 2022 - 44 = \bf 1978$