Question

ajutor va rog la a1. am nevoie​

Answer 1

Răspuns:

Explicație pas cu pas:

Answer 2

Explicație pas cu pas:

a)

$2 + \dfrac{1}{1} = 2 + 1 = 3 \\ 4 - \dfrac{1}{1} = 4 - 1 = 3$

$\implies I_{1} = \Big[3 ; 3\Big] = \Big \{3\Big \}$

$2 + \dfrac{1}{2} = 2\dfrac{1}{2} \\ 4 - \dfrac{1}{2} = 3\dfrac{1}{2}$

$\implies I_{2} = \Big[2\dfrac{1}{2} ; 3\dfrac{1}{2}\Big]$

$2 + \dfrac{1}{3} = 2\dfrac{1}{3} \\ 4 - \dfrac{1}{3} = 3\dfrac{2}{3}$

$\implies I_{3} = \Big[2\dfrac{1}{3} ; 3\dfrac{2}{3}\Big]$

$2 + \dfrac{1}{4} = 2\dfrac{1}{4} \\ 4 - \dfrac{1}{4} = 3\dfrac{3}{4}$

$\implies I_{4} = \Big[2\dfrac{1}{4} ; 3\dfrac{3}{4}\Big]$

=>

$I_{1} \cap I_{3} = \Big \{3\Big \} \cap \Big[2\dfrac{1}{3} ; 3\dfrac{2}{3}\Big] = \Big \{3\Big \}$

$I_{1} \cup I_{4} = \Big \{3\Big \} \cup \Big[2\dfrac{1}{4} ; 3\dfrac{3}{4}\Big] = \Big[2\dfrac{1}{4} ; 3\dfrac{3}{4}\Big]$

$\iff I_{1} \cup I_{4} = I_{4}$

$I_{2} - I_{3} = \Big[2\dfrac{1}{2} ; 3\dfrac{1}{2}\Big] - \Big[2\dfrac{1}{3} ; 3\dfrac{2}{3}\Big] = \O$

b)

$2 + \dfrac{1}{n} = 2\dfrac{1}{n}$

$4 - \dfrac{1}{n} = 3 + 1 - \dfrac{1}{n} = 3 + \dfrac{n - 1}{n} = 3\dfrac{n - 1}{n}$

$\implies I_{n} = \Big[2\dfrac{1}{n} ; 3\dfrac{n - 1}{n}\Big]$

$I_{n} \cap \mathbb {N} = \Big[2\dfrac{1}{n} ; 3\dfrac{n - 1}{n}\Big] \cap \mathbb {N} = \Big \{3\Big \}$

$\iff I_{n} \cap \mathbb {N} = I_{1}$