I am having trouble grasping the concept of an index family set. Here we have intersection and union, and from what I gather about the concept the answers should be as follows.
$a.$ empty set
$b.$ all real numbers set
$c.$ empty set
$d.$ (-1,3)
From the solutions the correct answer reads
$a.$ {0}
$b.$ all real numbers
$c.$ [0,1]
$d.$ (-1,3)
Any help would be appreciated, as it seems I'm having difficulty grasping the intersection vs. the union.
Best Answer
a. $0$ belongs to all sets $(-1/n,1/n)$, hence also to their intersection.
c. $[0,1]$ is contained in all sets $(-1/n,1+1/n)$, hence also to their intersection.
In both cases, it is clear they are "maximal", so it is exactly their intersection.
Ps. It seems you have difficulty only with intersections, but you understood unions. If this is the case, then use the identity $$ \bigcap_{i \in I}A_i=\left(\bigcup_{i \in I}A_i^c\right)^c, $$ and you are always back to unions.