$A_1 ∩ A_2 · · · ∩ A_n$is the set of all objects that are in all of the $A_j$’s

"For $$n$$ sets $$A_1 , . . . , A_n$$ , the union $$A_1 ∪ A_2 · · · ∪ A_n$$ is the set of all objects that are in at least one of the $$A_j$$ ’s, while the intersection $$A_1 ∩ A_2 · · · ∩ A_n$$ is the set of all objects that are in all of the $$A_j$$’s."

could somebody kindly help me understand the second part of the sentence "while the intersection $$A_1 ∩ A_2 · · · ∩ A_n$$ is the set of all objects that are in all of the $$A_j$$ ’s".

the first part which says that all objects that are in at least one of $$A_j$$ is easier to understand since $$A_j$$ is part of of the union set. however, all objects in the intersection set are in all of the $$A_j$$’s is confusing and i can't figure out a venn diagram to represent it.

• @Henry, thank you for the reply. i understand how the three circles works. however, the second part of the sentence says all objects in the intersection set are in all of $A_j$ sounds confusing as it like saying that $A_j$ is all the intersections Apr 14, 2022 at 11:25
• There is definite problem in using Venn's diagram, because we cannot put 4 circles in plane in such a way, that they form 16 areas. Apr 14, 2022 at 11:41
• If you find it easier, then all elements of the intersection are elements of each of the $A_j$. The intersection is a subset of each of the $A_j$ for $j\in \{1,2,\ldots,n\}$; indeed it is the largest such set which is a subset of each of the $A_j$ Apr 14, 2022 at 11:41
• @Henry, thanks this clears up the confusion of $A_j$ Apr 14, 2022 at 12:04
• @ManOnTheMoon Microsoft Paint - I am old (and old-fashioned) Apr 14, 2022 at 13:42

If you consider the classic Venn diagram of three circles, the union is the set of all the elements in any of the circles,

Union:

while the intersection is the set of all the elements in all of the circles i.e. the most central part of the Venn diagram

Intersection:

• thank you providing the venn diagram, it makes it easier to reference. How do one represent $A_j$ in your diagram? Apr 14, 2022 at 11:33
• @ManOnTheMoon In my second diagram the pink area is $A_1 \cap A_2 \cap A_3 = \bigcap\limits_{j=1}^3 A_j$. You could say it is $\{a: \forall j: \, a\in A_j\}$ Apr 14, 2022 at 11:38
• Similarly in the first diagram the purple area is $A_1 \cup A_2 \cup A_3 = \bigcup\limits_{j=1}^3 A_j$ and you could say it is $\{a: \exists j: \, a\in A_j\}$ Apr 14, 2022 at 12:06
• the purple area can also be represented as $A_1\cup A_2\cup A_3=A_1+A_2+A_3 -A_1\cap A_2-A_1\cap A_3-A_2\cap A_3+A_1\cap A_2\cap A_3$ right? Apr 14, 2022 at 12:30
• The use of $+$ and $-$ would need definition when applied to sets. But if the sets are finite and you are counting elements, or if you are applying a measure, then $|A_1\cup A_2\cup A_3|=|A_1|+|A_2|+|A_3| -|A_1\cap A_2|-|A_1\cap A_3|-|A_2\cap A_3|+|A_1\cap A_2\cap A_3|$ makes sense and is based on inclusion-exclusion Apr 14, 2022 at 13:14