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Exercitii-intervale in R

Astazi o sa efectuam cat mai multe exercitii intervale in R :
1) Se considera multimile:
<br /> \\A=\left\{x\in R|-2\leq x<3\right\}<br /> \\B=\left\{x\in R| -4<x\leq 1\right\}
a) Scrieti multimile A si B sub forma de interval
b) Determinati urmatoarele multimi
<br /> \\ C=\left\{x| x\in A\;\; si \;\;x\in B\right\}<br /> \\D=\left\{x| x\in A\;\; si \;\; x\in N^{*}\right\}<br /> \\E=\left\{x| x\in B\;\; si \;\; x\in Z^{*}\right\}<br /> \\F=\left\{x| x\in A\;\; si \;\; x\in B\right\}
Solutie:
<br /> \\A=[-2; 3)<br /> \\B=(-4;1]<br /> \\C=\left\{ -2; -1; 0; 1;\right\} sau ca interval [-2, 1]
\\D=\left\{1; 2\right\}<br /> \\E=\left\{-3; -2; -1; 1\right\}<br /> \\F=\left\{-2; -1; 0; 1\right\}
2) Calculati:
<br /> A\cup B; A\cap B; A-B; B-A<br /> \\A=\left\{x\in R|-1<\frac{3x+7}{2}\leq 11\right\}<br /> \\B=\left\{x\in R|-2\leq \frac{5x+9}{8}<3\right\}<br />
Solutie
Trebuie sa gasim multimile astfel daca inmultim
<br /> -1<\frac{3x+7}{2}\leq 11 |\cdot 2<br /> cu 2 o sa avem o inegalitate fara numitor, astfel obtinem:
<br /> \\-2<3x+7\leq 22 (-7)<br /> \\-2-7<3x+7-7\leq 22-7<br /> \\-9< 3x\leq 15 |:3<br /> \\-3<x\leq 5<br /> A=\left\{-2; -1; 0; 1; 2; 3; 4; 5\right\}<br />
Sau daca scrise sub forma de interval elementele multimii
<br /> A=(-3; 5]<br />
iar pentru multimea B luam
<br /> -2\leq \frac{5x+9}{8} \\-2\cdot 8\leq 5x+9<3\cdot 8<br /> \\-16\leq 5x+9< 24 (-9)<br /> \\-16-9\leq 5x+9-9<24-9<br /> \\-25\leq 5x \\-5\leq x< 3<br /> \\ B=\left\{-5; -4; -3; -2; -1; 0; 1; 2\right\},
iar daca scriem sub forma de interval obtinem:
<br /> B=[-5; 3)<br />
Calculam acum <br /> \\A\cup B=\left\{-5; -4; -3; -2; -1; 0; 1; 2; 3; 4; 5\right\}<br /> \\A\cap B=\left\{ -1; 0; 1; 2\right\}<br /> \\A-B=\left\{3; 4; 5\right\}<br /> \\B-A=\left\{-5; -4; -3;\right\}
iar daca scriem sub forma de interval obtinem:
<br /> \\A\cup B=[-5; 5]<br /> \\A\cap B=(-3; 3)<br /> \\A-B=[3; 5]<br /> \\B-A=[-5; -3]<br /> .
Deci trebuie sa rezovam cat mai multe exercitii intervale in R ca sa le intelegem mai bine.