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In the textbook I used on Probability and measure theory, a definition catches my eye:

If $I=\emptyset$, we define $\bigcup_{n\in I} A_n=\emptyset,\bigcap_{n\in I} A_n=\Omega$

I didn't draw attention to this definition initially, but this definition puzzles me a lot now. I don't get what it means to map an empty index set to a collection of a subset of $\Omega$. I also don't understand why the author would bother talking about the scenario where the index set is empty.

Asaf Karagila
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JoZ
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  • If you think of union as related to summation and intersection as related to multiplication then the empty set is like the union identity in the same way an empty sum is zero and an empty product is 1, so an empty intersection should be the "intersection identity" which should be the entire parent set. Unioning is constructive, so it starts with the empty set. Intersecting is reductive, so it starts with the parent set. Not an answer but just giving an intuitive way to think about it. – jdods Apr 15 '21 at 22:48
  • @OliverDiaz, ah... I deleted my previous comment, and will delete this one shortly... – paul garrett Apr 15 '21 at 22:59
  • Does this answer your question? Empty intersection and empty union – Chubby Chef Apr 15 '21 at 23:09

1 Answers1

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Sorry if my argument here is somewhat circular but I rather avoid getting into the guts of logic and set theory and stay within what one calls "naive set theory".

Recall that membership to a set $A$ can be described in terms of logical statements. Thais is, if $P$ is a proposition that can be either true or false (not both simultaneously), then $A=\{x\in \Omega: P(x)\}$ denotes the collection of elements in $\Omega$ that that make P(x) a TRUE statement.

For example, consider the set of all writes of the 19th century and let $A$ denote the set of writers that wrote a novel in Russian. Dostoyevsky satisfies the proposition wrote a novel in Russian, so Dostoyevsky is in $A$; Mark Twain (to my knowledge) did not ever write a novel in Russian, so Mark Twain is not in $A$.

No back to mathematics:

The intersection of a collection of sets $\{A_i:i\in I\}$ is defined as $$\bigcap_{i\in I}A_i=\{x\in\Omega: \text{for every $i$, if}\, i\in I,\,\text{then}\quad x\in A_i\}$$

When $I=\emptyset$, any $x\in\Omega$ makes the statement "if $i\in I$, then $x\in A_i$" a true statement, for the stamens $(i\in I)$ is always FALSE (on account that $I$ has no elements) in this case. Recall the table for the logic operation for $p\Rightarrow q$, which is equivalent to $\sim(q\wedge(\sim p))$ (negation of $q$ and no $p$):

  • "TRUE implies TRUE" is TRUE,

  • "TRUE implies FALSE" is FALSE,

  • "FALSE implies TRUE" is TRUE,

  • and "FALSE implies FLASE" is TRUE.

Similarly, the union of a collection of sets $\{A_i:i\in I\}$ is defined as $$ \bigcup_{i\in I}A_i=\{x\in \Omega: \text{there exists $i$ such that}\,\,i\in I,\,\text{and}\,\, x\in A_i\}$$ When $I$ is empty, then for every element $x$ of $\Omega$, the statement “there exist $i\in I$ and $x\in A_i$“ is FALSE (on account that $I$ has no elements. So in this case, $$ \bigcup_{i\in I}A_i=\emptyset$$ Here, recall the logic table for the logic operation $p \wedge q$. It is TRUE only when both statements $p$ and $q$ are TRUE.

Mittens
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  • I like to think of $P\to Q$ as $P \subset Q$ (subset). It's not a perfect analogy, but it helps me make sense of the truth values. FALSE implies TRUE just feels weird, but it is consistent with the subset idea. I'm just adding this here in case it helps any passers-by. – jdods Apr 15 '21 at 23:02
  • I was just adding a passing comment that I find helpful in understanding that particular logical implication. It's not a comment on your work. I can delete it if you think it is unhelpful or out of place. Let me clarify that the subset idea I pose has no rigorous connection to the logical $\Rightarrow$. It's just an informal idea. – jdods Apr 15 '21 at 23:14