Requesting a finite trivial example with this family of sets and union is associative to help me prove

Question

May I have a finite example where we define what a, $\cup$S, $F_a$, and C are? I think that would help for understanding this infinite proof. I don't see how this proof is true for finite examples of a, $\cup$S, $F_a$, and C.

Answer 1

Since $S$ is a system of sets, the notation $a\in \bigcup S$ is meaningful only if $a$ is a set; so $a$ is some element which must be in one or more of the sets in the system $S$. Then for each instance of $a$, $F_a$ is some set as well (which in general need not be in the system $S$), and $C$, ranging over the members of system $S$, takes on as values various sets, all of which are in $S$.

To give a finite example of these, let's work with the sets in $S$ being intervals on $\Bbb Z$ and take: $$ S = \{ [5n,5n+2] : n\in [1,3] \}\\ F_a = \{ ka^2 : k\in \Bbb Z \wedge (1\leq k \leq 4)\} $$ Then $$ a\in \bigcup S \Leftrightarrow a\in \{ 5,6,7,10,11,12,15,16,17\}\\ F_5 = \{25,50,75,100\}\\ F_6 = \{36,72,108,144\}\\ \cdots\\ F_{17} = \{ 289,578,867,1156 \} $$ and let's try out the union relation you are asked to prove:

It says that the union of all the elements of all the $F_a$s is equal to the union over the three sets in system $S$ (that is, $C$ takes on values $\{5,6,7\}$ then $\{10,11,12\}$ then $\{15,16,17\}$) of the sets formed by taking the union of all three $F_a$ for $a$ in the particular $C$ being considered.

That is, $$ \{25,50,75,100\} \cup \{36,72,108,144\} \cup \cdots \cup \{ 289,578,867,1156 \} \\ = (\{25,50,75,100\} \cup \{36,72,108,144\} \cup \{49,98,147,196\} ) \\ \cup (\{100,200,300,400\} \cup \{121,242,363,484\} \cup \{144,288,432,576\} ) \\ \cup (\{225,450,675,900\} \cup \{256,512,768,1024\} \cup \{ 289,578,867,1156 \}) $$

This is pretty trivially true; the gist of the problem is to prove it is true in general, including for infinite sets and infinite systems of sets without using any axioms that are unavailable in your logic/set theory.