Use De Morgan's Laws to simplify the following sets
Simplify the following sets:
$$ℝ\setminus \bigcap\limits_{n=1}^∞ (-1/n,1/n)\tag1$$
$$\bigcup\limits_{n=1}^∞ (ℝ\setminus[1/n,2+1/n])\tag2$$
For the first problem, I used De Morgan's law, and it equals to $$\bigcup\limits_{n=1}^∞ (ℝ\setminus[-1/n,1/n])\tag2$$ and that is $$\bigcup\limits_{n=1}^∞ ([-∞,-1/n]\bigcup[1/n,+∞])\tag2$$ Well I think that is the set ℝ. But I'm not sure I can't prove it.
For the second problem, using De Morgan's law, it equals
$$ℝ\setminus \bigcap\limits_{n=1}^∞ (1/n,2+1/n)\tag1$$
I don't know what $$\bigcap\limits_{n=1}^∞ (1/n,2+1/n)\tag1$$ is, but I think it's [0,2]. But I can't prove it either way. I'm really sure this question really wants me to simply to an interval on the real number line.
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$\begingroup$HINT: $A\setminus B = A\cap B^c$
Use this then apply DeMorgan's Laws and Distribution, to simplify the given sets.
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