Thread: [SOLVED] Finding the union and intersection of indexed sets.

1. Originally Posted by Plato
Here is a strong hint on the intersection in #4.
$\left( {\forall x \in (0,1)} \right)\left[ { - \frac{1}
{x} < - 1 < 1 < \frac{1}
{x}} \right]
$
So based on that interval, it looks like (-1, 1) is the intersection.
Did I get it? or not

Here are my revised answers after all the help I got:

(1)
= (-1, 1)
= [0, 1)

--------------------------

(2)
= Q
=
--------------------------

(3)

--------------------------
(4)
= (- $\infty$, $\infty$)
= (-1,1)
--------------------------

(5)
= [0,1)
= {0}

--------------------------

(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!

3. Originally Posted by vincisonfire
The notation R\Q means all real numbers EXCEPT rational. You thus know that z in never rational.
You know that for set An, z is smaller than n that is a natural number.
Eg A10 contains all real number smaller than 10 that are not rational.
I would say that union contains R\Q and the intersection should be the empty set.
does this refer to #5 or #6? They both have 'z' but #5 has the $A_n$ and #6 has $A_z$

4. The union in #5 ought to be $\left( {0,1} \right) \cap \left( {\mathbb{R}\backslash \mathbb{Q}} \right) = \left\{ {x \in :\left( {\mathbb{R}\backslash \mathbb{Q}} \right):0 < x < 1} \right\}$.
Do you see that in #5, $\left( {\forall n \in \mathbb{N}} \right)\left[ {A_n \subseteq \left( {\mathbb{R}\backslash \mathbb{Q}} \right)} \right]$?
Thus, the intersection is $\emptyset$.

On the other hand, in #6, $\left( {\forall z \in \left( {\mathbb{R}\backslash \mathbb{Q}} \right)} \right)\left[ {A_z \subseteq \mathbb{N}} \right]$.

5. Rethinking

Originally Posted by Plato
The union in #5 ought to be $\left( {0,1} \right) \cap \left( {\mathbb{R}\backslash \mathbb{Q}} \right) = \left\{ {x \in :\left( {\mathbb{R}\backslash \mathbb{Q}} \right):0 < x < 1} \right\}$.
Do you see that in #5, $\left( {\forall n \in \mathbb{N}} \right)\left[ {A_n \subseteq \left( {\mathbb{R}\backslash \mathbb{Q}} \right)} \right]$?
Thus, the intersection is $\emptyset$.

On the other hand, in #6, $\left( {\forall z \in \left( {\mathbb{R}\backslash \mathbb{Q}} \right)} \right)\left[ {A_z \subseteq \mathbb{N}} \right]$.
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 $A_z$ will be all irrationals less than all the naturals.