Here are my revised answers after all the help I got:
(1)
= (-1, 1)
= [0, 1)
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(2)
= Q
=
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(3)
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(4)
= (-$\displaystyle \infty$, $\displaystyle \infty$)
= (-1,1)
--------------------------
(5)
= [0,1)
= {0}
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(6)
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(Do they all look correct now? ) This is for a boardwork in class assignment (we present one of these in front of the class ).
We don't know which problem we'll present until just minutes before we present, so I really wanted to make sure I got them all okay so I can do well in the boardwork presentation in class. Thanks!
The union in #5 ought to be $\displaystyle \left( {0,1} \right) \cap \left( {\mathbb{R}\backslash \mathbb{Q}} \right) = \left\{ {x \in :\left( {\mathbb{R}\backslash \mathbb{Q}} \right):0 < x < 1} \right\}$.
Do you see that in #5, $\displaystyle \left( {\forall n \in \mathbb{N}} \right)\left[ {A_n \subseteq \left( {\mathbb{R}\backslash \mathbb{Q}} \right)} \right]$?
Thus, the intersection is $\displaystyle \emptyset$.
On the other hand, in #6, $\displaystyle \left( {\forall z \in \left( {\mathbb{R}\backslash \mathbb{Q}} \right)} \right)\left[ {A_z \subseteq \mathbb{N}} \right]$.
Thus you need to redo your answers.
So I guess vince's answer was referring to #5 then. So are all the rest correct? (besides my #5 and 6).
For #6, the notation is throwing me off. I'm still confused how to read it. I see that z is any irrational number and the sets in the $\displaystyle A_z$ will be all irrationals less than all the naturals.