Question

Determinati cardinalul multimii :
B = {x€Z | 6/2x-1 € Z}

€ = apartine
/ = linie de fractie

Answer 1

$B=\left \{ x\:\in\:\mathbb{Z} \:|\:\:\frac{6}{2x-1}\:\in\:\mathbb{Z}\right \}$

$\frac{6}{2x - 1} \: \in \: \mathbb{Z} = > 2x - 1 \: \in \: D_{6}$

$D_{6}=\left \{ \pm1,\pm2,\pm3,\pm6 \right \}$

$1)2x - 1 = - 1$

$2x = - 1 + 1$

$2x = 0$

$x = 0 \: \in \: \mathbb{Z}$

$2)2x - 1 = 1$

$2x = 1 + 1$

$2x = 2 \: | \div 2$

$x = 1 \: \in \: \mathbb{Z}$

$3)2x - 1 = - 2$

$2x = - 2 + 1$

$2x = - 1$

$x = - \frac{1}{2} \: \notin \: \mathbb{Z}$

$4)2x - 1 = 2$

$2x = 2 + 1$

$2x = 3$

$x = \frac{3}{2} \: \notin \: \mathbb{Z}$

$5)2x - 1 = - 3$

$2x = - 3 + 1$

$2x = - 2 \: | \div 2$

$x = - 1 \: \in \: \mathbb{Z}$

$6)2x - 1 = 3$

$2x = 3 + 1$

$2x = 4 \: | \div 2$

$x = 2 \: \in \: \mathbb{Z}$

$7)2x - 1 = - 6$

$2x = - 6 + 1$

$2x = - 5$

$x = - \frac{5}{2} \: \notin \: \mathbb{Z}$

$8)2x - 1 = 6$

$2x = 6 + 1$

$2x = 7$

$x = \frac{7}{2} \: \notin \: \mathbb{Z}$

$= > B=\left \{ -1,0,1,2\right \}$

$= > cardB = 4$