Question

Calculati multimea B={x apartine lui Z|x-2/x+2 apartine lui Z}

Answer 1

X-2/x+2 E Z
Rezulta x-2|x+2
știm ca x-2|x-2 /•(-1)
Rezulta x-2|x+2
Iar. x-2|-x+2 +
---------------------------------
x-2|x+2-x+2
x-2|4
x-2 E div 4={-4,-2,-1,1,2,4}
x-2=-4,x=-2
x-2=-2,x=0
x-2=-1,x=1
x-2=1,x=3
x-2=2,x=4
x-2=4,x=6
Rezulta B={-2,0,1,3,4,6}

Answer 2

Determinați  elementele mulțimii B = {x∈ℤ| (x-2)/(x+2)∈ℤ}


[tex]\it \dfrac{x-2}{x+2} \in\mathbb{Z} \Leftrightarrow \dfrac{x+2-4}{x+2} \in\mathbb{Z} \Leftrightarrow \dfrac{x+2}{x+2} -\dfrac{4}{x+2}\in\mathbb{Z} \Leftrightarrow 1- \dfrac{4}{x+2} \in\mathbb{Z} \\\;\\ \\\;\\ \Leftrightarrow \dfrac{4}{x+2} \in\mathbb{Z} \Leftrightarrow (x+2)|4 \Leftrightarrow x+2\in D_4 \Leftrightarrow x+2 \in\{\pm1,\ \pm2,\ \pm4\} [/tex]


$\it x+2 \in \{-4,\ -2,\ -1,\ 1,\ 2,\ 4\}|_{-2} \Rightarrow x \in \{-6,\ -4,\ -3,\ -1,\ 0,\ 2\}$


Prin urmare, mulțimea cerută este :

B = {-6,  -4,  -3,  -1,  0,  2}