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I have read that mathematicians like to be lazy, but I find this really disturbing as there is clearly an error somewhere (with my over-significance of the words, the wiki, or the AMS notes)

Background

These lead me to my guesses/thoughts.

Here, the power set $\mathscr{P}(S)$ is defined as a "family of sets over $S$". Similarly, "any family of subsets of $S$ is itself a subset..." is written on the same page.

What conflicts is this (download the pdf, pg. 93-94), which says "A family of subsets of a set $X$ consists of ..." and "... be a family of subsets of $X$".

I know that I have definitely heard a union defined as "...$\bigcup$ over $S$..."

Questions

  1. Does the difference between the words "of" and "over" have a significance in notation, proofs, or work?

  2. What do the words mean? And, if there is a difference, please explain in layman's terms if possible as the difference might too mathematical/rigorous for an introduction to Set Theory.

  3. Extra: I am really struggling to understand the notation on pg. 94 about the rigorous definition of a union and intersection, and they look exactly the same (no joke) here. You might have to download the pdf though. Any explanation would be greatly appreciated!

My thoughts

Referencing the background, I assume a family of sets is an operation, and performing operations on a set is written as some operation over some set. Likewise, of is referencing the set or some object it is pointing to.

This thought failed immediately because families of sets would not be an operation, but the guess for of might indicate seems strong.

sreysus
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  • I'm sorry, I'm not going to download pdfs from elsewhere – questions here should be self-contained insofar as possible. But at least I can say this: suppose $S$ and $T$ are sets, and suppose that for every $s$ in $S$ there is some subset of $T$, a subset we'll call $A_s$. Then we can write $\cup_{s\in S}A_s$, and call it a union over $S$, the same way that $\sum_{n=1}^{17}f(n)$ is a sum over ${1,2,\dotsc,17}$, or $\int_0^1f(x),dx$ is an integral over the interval $[0,1]$. – Gerry Myerson Sep 14 '23 at 03:09
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    "Over" and "of" don't have standard mathematical meanings by themselves - we usually directly define larger phrases like "vector space over [some field]" or "subgroup of [some group]". – Karl Sep 14 '23 at 06:10

1 Answers1

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To answer your questions:

  1. Does the difference between the words "of" and "over" have a significance in notation, proofs, or work?

Not at all, regardless of how we can define the power set intuitively using "natural language" (the language we use everyday), we do have a formal (most unambiguous as possible) definition. Look, the existence of the power set is an axiom of set theory:

Axiom (existence of power set). For any set $X$, there is a set $S$ such that $A \in S$ if and only if $A \subset X$.

Definition (power set of a set X). Given a set $X$, we call the set $S$ (given by the axiom) the power set of $X$, or the power set over $X$, and it is denoted by the symbol "$P(X)$".

What does this axiom/definition say? it says that if you have any set $X$, you can automatically asure that you have another set $"P(X)"$, that has the property that any subset of $X$ is an element of $P(X)$.

Example. Consider the set $X = \{1,2,3\}$. By the axiom of power set, there is a set $P(X)$ such that $A \in P(X)$ if and only if $A \subset X$. Which are those "$A$" subsets of $X$?, these are:

  1. $\emptyset \subset \{1,2,3\}$ (this is true for any set $X$, if you have doubts about this, check this question)
  2. $\{1\} \subset \{1,2,3\}$
  3. $\{2\} \subset \{1,2,3\}$
  4. $\{3\} \subset \{1,2,3\}$
  5. $\{1,2\} \subset \{1,2,3\}$
  6. $\{1,3\} \subset \{1,2,3\}$
  7. $\{2,3\} \subset \{1,2,3\}$
  8. $\{1,2,3\} \subset \{1,2,3\}$

Since these are all the posibilites, we have that $$P(\{1,2,3\}) = \{ \emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$$

This way we have indeed that:

  1. $\emptyset \in P(\{1,2,3\})$
  2. $\{1\} \in P(\{1,2,3\})$
  3. $\{2\} \in P(\{1,2,3\})$
  4. $\{3\} \in P(\{1,2,3\})$
  5. $\{1,2\} \in P(\{1,2,3\})$
  6. $\{1,3\} \in P(\{1,2,3\})$
  7. $\{2,3\} \in P(\{1,2,3\})$
  8. $\{1,2,3\} \in P(\{1,2,3\})$

Fun fact (theorem). If $X$ has $n$ elements, then $P(X)$ has $2^n$ elements.

2. What do the words mean? And, if there is a difference, please explain in layman's terms if possible as the difference might too mathematical/rigorous for an introduction to Set Theory.

I would suggest that, in general, any time you come across a mathematical definition written in "plain english" that you can't grasp or understand, instead of trying to interpret what the author meant, go look at the formal definition (there is always one!), this definition should take out any ambiguity in the "plain english" definition (for me this is one of the beauties of mathematics). So, my recommendation is to not think too much on the "intuitive" definitions and the words "over" or "of", just realize that these authors are talking about the same mathematical object defined in the previous paragraphs.

3. I am really struggling to understand the notation on pg. 94 about the rigorous definition of a union and intersection, and they look exactly the same (no joke) here.

On page 94 of the pdf you provide we see the following:

"Definition. Let $X$ be a set. (1) Let $\{U_i\}_{i \in I}$ be a family of subsets of $X$. We write

$$\bigcup_{i \in I} U_i = \{x \in X \; | \textbf{ there exists an }i \in I \textbf{ with } x \in \color{red}{X_i}\} \quad \text{(Union)} $$

$$\bigcap_{i \in I} U_i = \{x \in X \; | \textbf{ for every }i \in I \textbf{ we have } x \in \color{red}{X_i}\} \quad \text{(Intersection)}".$$

First of all this definition is wrong, we don't know what are those $X_i$'s, they confused $X's$ for for $U's$, the correct definition is this one:

Definition. Let $X$ be a set. Let $\{U_i\}_{i \in I}$ be a family of subsets of $X$ (that is, $U_i \subset X$ for all $i \in I$). We write

$$\bigcup_{i \in I} U_i = \{x \in X \; | \textbf{ there exists an }i \in I \textbf{ with } x \in \color{blue}{U_i}\} \quad \text{(Union)} $$

$$\bigcap_{i \in I} U_i = \{x \in X \; | \textbf{ for every }i \in I \textbf{ we have } x \in \color{blue}{U_i}\} \quad \text{(Intersection)}.$$

And they are not the same, why?, let's consider the set

$$X = P(\{1,2,3\}) = \{ \emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\},$$

We need a family of subsets of $X$, let's consider

$U_1 = \{\{1\}\}$,

$U_2 = \{\{1\},\{1,2\}\}$,

$U_3 = \{\{1\},\{1,3\}\}$ and

$U_4 = \{\{1\},\{2\}\}$,

we see that $U_i \subset X$ for all $i \in \{1,2,3,4\} = I$,

One one hand what set is $\bigcup_{i \in I} U_i$?

By the definition, $\alpha \in \bigcup_{i \in I} U_i$ if and only if there is $i \in I$ such that $\alpha \in U_i$.

For example, is $1 \in \bigcup_{i \in I} U_i?$, no, because $1 \notin U_i$ for all $i \in I$, in other words, there is no particular $i \in I$ such that $1 \in U_i$.

Is $\{2\} \in \bigcup_{i \in I} U_i$?, yes, because there is $i = 4$ that fulfills that $\{2\} \in U_4 = \{\{1\},\{2\}\}$.

This way you can prove that

$$\bigcup_{i \in I} U_i = \{\{1\},\{2\},\{1,2\},\{1,3\}\}.$$

On the other hand what set is $\bigcap_{i \in I} U_i$?

By the definition, $\alpha \in \bigcap_{i \in I} U_i$ if and only if $\alpha$ fulfills that $\alpha \in U_i$ for all $i \in I$.

For example, is $\{2\} \in \bigcup_{i \in I} U_i?$, no, because, although $\{2\} \in U_4$, it also happens that $\{2\} \notin U_i$ for $i \in \{1,2,3\}$, that is, $\{2\}$ is not in every $U_i$ as the definition asks.

Is $\{1\} \in \bigcap_{i \in I} U_i$?, yes, because $\{1\}$ is an element of every $U_i$, that is $\{1\} \in U_i$ for all $i \in \{1,2,3,4\}$.

This way you can prove that

$\bigcap_{i \in I} U_i = \{\{1\}\}$.

César VB
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