Union, intersection and complementary of intervals

Calculate the following sets, and say if they are intervals or not, and classify them,

  1. $\overline{(1,8)\cap[-2,3]}$
  2. $\overline{[\sqrt{5},9]}\cup\overline{(-2,\dfrac{\sqrt{2}}{3})}$
  3. $\overline{(-\infty,-\dfrac{5}{7})\cup\overline{(-4,+\infty)}}$
  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: $$\overline{[1,3)}=(-\infty,1)\cup[3,+\infty)$$

  1. 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 $\mathbb{R}.$

  1. 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: $$[-\dfrac{5}{7},+\infty)\cap(-4,+\infty)=[-\dfrac{5}{7},+\infty) $$

  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.
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