# Could someone explain what A is, and how I could work my way to finishing this problem?

What can you say about $$\bigcap_{i\geq 1}A_i$$ if for each $$i$$ we have $$A\subset A_i$$?

(A) $$A\subset \bigcap_{i\geq 1}A_i$$
(B) $$\bigcap_{i\geq 1}A_i\subset A$$
(C) There exists $$i$$ s.t. $$A_i\subset A$$

What exactly does $$A$$ mean - is that the union of all $$A_i$$ or the intersection of all $$A_i$$, or neither?

• $A$ is just another set. Feb 11, 2022 at 0:36
• $A$ is any set that is a subset of each of the $A_i$. In the future, posts the actual problem, not an image. Use mathjax: math.meta.stackexchange.com/questions/5020/… Feb 11, 2022 at 0:36
• Right, but how does that A pertain to Ai? Like, if say, we assume Ai = [3,i], what would A be equal to? or are A and Ai separate entities? ^^^ above would A be the intersection of the subsets then? Because if A C Ai for all i values, wouldn't it have to satisfy all i values, and thus be an intersection of them all?
– user1021735
Feb 11, 2022 at 0:37
• $A \subset A_1, A \subset A_2, A \subset A_3, \cdots.$ Now, compare this inference against each of the $3$ offered choices, and see if the inference corresponds to one of them. For example, if $A \subset A_1$ and $A \subset A_2$, does this imply that $A \subset \left[A_1 \cap A_2\right]$? Feb 11, 2022 at 0:39
• Oh so in that case, it would be b correct? because if A is a subset of all of them then A is a subset of the intersection of them? Sorry guys I am really new to set notation and all
– user1021735
Feb 11, 2022 at 0:41

From what I can understand from the comments on this post $$A$$ is a subset of each $$A_i$$. I’ll take each of a, b and c and try to explain to my best knowledge:
a) It is true, because the intersection of all $$A_i$$’s will at the very least be $$A$$ itself since it is a subset of all of them. That means any element in $$A$$ is in $$A_1,A_2$$ and so on.
b) This one can’t be true. Let’s assume any $$a\in \displaystyle\cap_{i\geq 1} A_i$$, for which $$a \notin A$$, then that statement is not true. Plus at the very least, in accordance to what I mentioned in (a) you’d need to have an equality included since that’s the only way $$\displaystyle\cap_{i\geq 1}A_i \subseteq A$$ would be true.
(c) That is also untrue provided that $$A$$ doesn’t coincide with $$A_i$$ since we’re using “$$A\subset A_i$$” and not “$$A\subseteq A_i$$”.