Problems from Union, intersection and complementary of intervals

Development:

1. We calculate first the intersection, and then we will calculate the complementary of the result given. Observing the endpoints of the given intervals, we have the following order: $$-2 < 1 < 3 < 8$$

   So, we know that the values between $$1$$ and $$3$$ belong to both intervals, and therefore, they belong to the intersection. So the result of the intersection is: $$$(1,8)\cap[-2,3]=[1,3)$$$ Now let's calculate the complementary of this interval: $$$\overline{[1,3)}=(-\infty,1)\cup[3,+\infty)$$$

2. We calculate first the complementary: $$$\overline{[\sqrt{5},9]}=(-\infty,\sqrt{5})\cup(9,+\infty)$$$ $$$\overline{(-2,\dfrac{\sqrt{2}}{3})}=(-\infty,-2]\cup[\dfrac{\sqrt{2}}{3},+\infty)$$$

   Then, as $$\dfrac{\sqrt{2}}{3} < \sqrt{5}$$, the union is the entire $$\mathbb{R}.$$

3. We have: $$\overline{(-\infty,-\dfrac{5}{7})\cup\overline{(-4,+\infty)}}= \overline{(-\infty,-\dfrac{5}{7})} \cap \overline{\overline{(-4,+\infty)}}$$

   But, as the complementary of the complementary is the same set, we have:

   $$\overline{(-\infty,-\dfrac{5}{7})} \cap \overline{\overline{(-4,+\infty)}}=\overline{(-\infty,-\dfrac{5}{7})} \cap (-4,+\infty)$$

   If we calculate the complementary: $$\overline{(-\infty,-\dfrac{5}{7})}=[-\dfrac{5}{7},+\infty)$$

   So finally, we calculate the intersection: $$$[-\dfrac{5}{7},+\infty)\cap(-4,+\infty)=[-\dfrac{5}{7},+\infty) $$$

Solution:

1. $$(-\infty,1)\cup[3,+\infty):$$ it is not an interval, since it is the union of two intervals, both unbounded and one open and the other closed.

2. $$\mathbb{R}=(-\infty, +\infty):$$ it is an unbounded interval.

3. $$[-\dfrac{5}{7},+\infty):$$ it is a closed interval, and unbounded above.

Hide solution and development