Question

aflați elementele următoarelor mulțimi:
A={x€N|8/x. €N}
B={x€N|12/x+1€ N}
C={x€N|21/2x-1 € N}
D={x€N|x+10/x+2 €N}
E={x€N|2x+11/x+2 €N}
F={x€N|3x+15/2x+3 €N}

plzzz. Dau coroană. Nu toate câte puteți.

Answer 1

Deci pt inceput trebuie sa stii ca la toate multimile de mai sus au toate elementele cu nr naturale si trebuie sa respecte impartirea respectiva
A = (0,1,2,4,8)  ex: 8/1 = 8; 8/2=4.
B = (0,1,2,3,5,11)
C = (1,2,4,11)
D = (0,2,6)
E = (5)
F = (0,2,9)

Answer 2

B={x€N|12/x+1€ N}

[tex]\it \dfrac{12}{x+1} \in\mathbb{N} \Rightarrow x+1|12 \Rightarrow x+1\in D_{12} \Rightarrow x+1 \in \{1,2, 3, 4, 6, 12\}|_{-1} \\\;\\ \\\;\\ \Rightarrow x \in \{0,\ 1,\ 2,\ 3,\ 5,\ 11\}[/tex]

B = {0,  1,  2,  3,  5,  11}
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E={x€N|2x+11/x+2 €N}

[tex]\it \dfrac{2x+11}{x+2} \in \mathbb{N} \Rightarrow x+2|2x+11 \ \ \ \ (1) \\\;\\ \\\;\\ Dar,\ \ x+2|x+2 \Rightarrow x+2| 2(x+2) \Rightarrow x+2| 2x+4 \ \ \ \ (2) \\\;\\ \\\;\\ (1),\ (2) \Rightarrow x+2| 2x+11-2x-4 \Rightarrow x+2| 7 \Rightarrow x+2\in\{1,7\} |_{-2} [/tex]

[tex]\it \Rightarrow x \in \{-1,\ 5\} \\\;\\ Dar,\ x\in \mathbb{N}, \ deci\ \ x=7. \\\;\\ E=\{5\} [/tex]