Proof of a set union/intersection

[bargaj]

Hello!
Lately, I've been struggling with this assignment. (angle brackets represent closed interval)



I figured out that:

a)
union = R
intersection = {0}
b)
union = (0, 2)
intersection = {1}

I asked my prof about this and she explained to me that it should be shown that if a set is an intersection of sets, then it belongs to each of those sets and, conversely, nothing else belongs to the intersection, so every other element does not belong to at least one of those sets. But I don't really know how to interpret this or where to even start. (normally, when proving the equality of two sets, I would try to prove that A⊆B and B⊆A, but I don't see how that's applicable here).

Thank you for your help!

 

[blamocur]

The way I'd go about the intersections is to A) prove that your answer belongs to the intersection, and B) prove that nothing else does.
Does this help?

 

[lev888]

if a set is an intersection of sets, then it belongs to each of those sets and, conversely, nothing else belongs to the intersection, so every other element does not belong to at least one of those sets.

How I would do it: pick an arbitrary point in the intersection set and prove that it belongs to all sets. Similar approach in the other direction.

E.g.
a) Intersection:
0 belongs to all sets specified as interval [-t, t] where t>0.
Let's pick a point u>0. Whatever u is, we can set t=u/2. This will define an interval to which u does not belong.

 

[pka]

Frankly I have never seen [imath]\left<-t,t\right>[/imath] used as an interval notation.
I assume that it is what most of the mathematical world means as closed interval [imath]\left[-t,t\right][/imath]
For the union in part a) suppose that [imath]x\in\mathcal{R}[/imath] then suppose that [imath]t=|x|[/imath] then
[imath]x\in\left[-t,t\right]\;T = \left( {0,\infty } \right) \Rightarrow x \in \bigcup\limits_{t \in T} {\left[ { - t,t} \right]} [/imath]
For the intersection in part a) take note that [imath]\left( {\forall t > 0} \right)\left[ {0 \in \left[ { - t,t} \right]} \right][/imath]
Can you do part b) ? Please show us.

[imath][/imath][imath][/imath][imath][/imath][imath][/imath][imath][/imath]