Question

24. Fie mulţimile A = {xe N | 3 < 2[2(4x-3)+13]-27 <115}, B = {x e N | 13 <2[(3(2x-3)+17]-27<
<97}.
a) Determinaţi mulţimile A şi B.
b) Calculaţi AUB, AnB, A-B şi B-A.

Answer 1

Explicație pas cu pas:

$3 < 2[2(4x-3)+13]-27 <115\ \ \Big|( + 27)$

$30 < 2[2(4x-3)+13] < 142\ \ \Big|(:2)$

$15 < 2(4x-3)+13 < 71\ \ \Big|( - 13)$

$2 < 2(4x-3) < 58\ \ \Big|(:2)$

$1 < 4x-3 < 29\ \ \Big|( + 3)$

$4 < 4x < 32\ \ \Big|(:4)$

$1 < x < 8$

=> A = {2, 3, 4, 5, 6, 7}

$13 < 2[3(2x - 3) + 17] - 27 < 97\ \ \Big|( + 27)$

$40 < 2[3(2x - 3) + 17] < 124\ \ \Big|(:2)$

$20 < 3(2x - 3) + 17 < 62\ \ \Big|( - 17)$

$3 < 3(2x - 3) < 45\ \ \Big|(:3)$

$1 < 2x - 3 < 15\ \ \Big|( + 3)$

$4 < 2x < 18\ \ \Big|(:2)$

$2 < x < 9$

=> B = {3, 4, 5, 6, 7, 8}

.

A ∪ B = {2, 3, 4, 5, 6, 7, 8}

A ∩ B = {3, 4, 5, 6, 7}

A - B = {2}

B - A = {8}