Order of evaluation in conditions in set theory

Question

Halmos, in Naive Set Theory, on page 19, provides a definition of intersection restricted to subsets of $E$, where $C$ is the collection of the sets intersected. The point is to allow the case where $C$ is $\emptyset$, which with this definition of intersection gives $E$ as the result.

$\{x \in E: x \in X$ for every $X$ in $C\}$

My problem lies in interpreting the sentence. I wanted to read it as:

"Elements x in E, given that: Element x is in X for every X in C"

My brain, tuned by a number of popular programming languages, wants to evaluate the terms in the condition reading from left to right. And clearly, no element $x$ will be in any $X$ if $C$ is $\emptyset$, and if the condition is evaluated to false, $E$ will not be the result of the intersection.

After struggling for a while, I figured that I had to read the sentence as:

"Elements x in E, given that: For all X that are in C, x is in all of them"

The for part of the condition has to be the pivotal one. It has to be the first term you evaluate. In analogy with common programming languages.

Questions:

1. Is my new reading and conclusion correct?
2. How does one learn the order of evaluation in set theoretic expressions?

Edit: Corrected after discussion with coldnumber.

Edit 2: Upon rereading the previous chapter, I've found that Halmos actually explains his "for every". The condition "$x \in X$ for every $X$ in $C$" actually means "for all $X$ (if $X \in C$, then $x \in X$)" -- which seems to give an unambiguous order of evaluation.

Answer 1

The first highlighted statement is mixing English reading of a statement with mathematical notation. The clue for that is that it had to say "for every..."

WHhen you do this, it becomes a bit less clear as to what in order you have to "resolve" the instructions. In this case, because the statement is so short and simple, one has hopes of getting it right.

It would be cleaner if you expressed it in math notation in the first place:

$$S(C) =\{ x \in E : \left( \forall X \subset C : x \in X \right)\}$$

Mpw it becomes clear that to test whether $x$ is in $S(C)$ you have to verify that $x\in E$ and then "try" every possible subset of $C$ and decide of $x$ is in that subset -- and if any of those fail, $x$ is not in $S(C)$.

Now it is very clear that $$S(\emptyset) = E$$

Answer 2

maybe think of it this way (remembering that your sets $X$ belong to some universe which is not to be identified with the collection $C$). to avoid confusion i use the bound variable $y$ for the universally quantified statement (in place of your $X$): $$ \{x: (x \in E) \land \forall y (y \in C \Rightarrow x \in y)\} $$

Answer 3

Looking at the book, I see that $C$ is a collection of subsets of $E$, and the definition in your question is the intersection of all the elements of $C$. Note that a subset $X$ of $E$ is not a subset of $C$; it is an element of $C$.

Your initial reading is correct. The set $N=\{x \in E: x \in X$ for all $X \in C\}$ contains the elements of $E$ that are in every $X \subset E$ that is an element of $C$.

This means that if, for example, if $E=\{a,b\}$ and $C = \{\{a\}, E\}$, then $N = \{a\}$

On the other hand, if $E=\{a,b\}$ and $C = \{\varnothing, \{a\}, E\}$, then $N = \varnothing$, because $\varnothing$ contains no element of $E$.

Or if $C = \{\{b\} \{a\}, E\}$, then $N =\varnothing$, because there is no element of $E$ that is in both $\{a\}$ and $\{b\}$.

In general, $N =\varnothing$ when $C$ contains subsets of $E$ that are disjoint.

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EDIT: Now, if $C = \varnothing$, and we look for $N = \{x \in E: x \in X \;\forall X \in C\}$, it follows that $N=E$, because to find an element $x$ of $E$ that does not satisfy the condition "$x \in X \; \forall X \in \varnothing$" we would have to find an element $X \in \varnothing$ that does not contain $x$, which is impossible, so every $x \in E$ satisfies the condition.