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- TL;DR Summary
- Is there a better way to approach thinking about equality with infinite sets?

I am reading an abstract algebra textbook and enjoying it. I am working through preliminaries some more to refine my knowledge on proofs with sets before really digging in. I understand that if

$$X \subseteq Y$$

and

$$ Y \subseteq X$$

Then

$$ X = Y$$

This makes sense to me. However, the following problem while simple caused problems with brain trying to establish a "bound" to make things make sense.

Suppose we have the following two sets and wish to prove them equal:

$$C = \\{x | x = 3r - 1, r \in Z\\}$$

$$D = \\{x | x = 3s + 2, s \in Z \\}$$

Then the proof can go as follows which I hamfisted using algebra:

Note that $x = 3r - 1$ can be rewritten $x=3(s+1)-1 = 3s+2$. Since this is exactly the set $D$ for every s in Z, D will contain a c in C at s+1. Then $$C \subseteq D$$.

Conversely, note that $x = 3s + 2$ can be rewritten $x = 3(r-1) + 2 = 3r - 1$. Then for every r in Z, C will contain a d in D at r - 1. So $D \subseteq C$, and $C = D$.

The arithmetic follows. The two equations are equal at $r -1$ and $s+1$ for any $r$ and and $s$. I drew up a table and indeed the pattern is obvious. It is "off by one" so to speak, as indicated by the solutions. But this "off by one" property is exactly what is causing me confusion.

There are two contradictory things in my head right now:

1. If we fix $s$ and $r$ to "run" for the same length it will always be off by one. So this tells me the sets aren't

2. However, if we see that we can

So I think the way to understand this is that the infinite sets are not really "built" incrementally, but rather by the set builder notation come into existing as a whole (if that makes sense). Then, no matter the case, we can always choose an $s$ and $r$ in these infinite sets proving that they're equal. But if I think about it like a person who might play with the sets first to see if the conjecture holds you will find it's always off by one. These two contradictory things I am trying to resolve and I can't split them enough in my head. Can anyone provide a better explanation?

$$X \subseteq Y$$

and

$$ Y \subseteq X$$

Then

$$ X = Y$$

This makes sense to me. However, the following problem while simple caused problems with brain trying to establish a "bound" to make things make sense.

Suppose we have the following two sets and wish to prove them equal:

$$C = \\{x | x = 3r - 1, r \in Z\\}$$

$$D = \\{x | x = 3s + 2, s \in Z \\}$$

Then the proof can go as follows which I hamfisted using algebra:

Note that $x = 3r - 1$ can be rewritten $x=3(s+1)-1 = 3s+2$. Since this is exactly the set $D$ for every s in Z, D will contain a c in C at s+1. Then $$C \subseteq D$$.

Conversely, note that $x = 3s + 2$ can be rewritten $x = 3(r-1) + 2 = 3r - 1$. Then for every r in Z, C will contain a d in D at r - 1. So $D \subseteq C$, and $C = D$.

The arithmetic follows. The two equations are equal at $r -1$ and $s+1$ for any $r$ and and $s$. I drew up a table and indeed the pattern is obvious. It is "off by one" so to speak, as indicated by the solutions. But this "off by one" property is exactly what is causing me confusion.

There are two contradictory things in my head right now:

1. If we fix $s$ and $r$ to "run" for the same length it will always be off by one. So this tells me the sets aren't

*always*equal.2. However, if we see that we can

*choose*$s$ and $r$ we can always choose $s$ and $r$ such that they produce the same value. In fact, this comes directly from the arithmetic.So I think the way to understand this is that the infinite sets are not really "built" incrementally, but rather by the set builder notation come into existing as a whole (if that makes sense). Then, no matter the case, we can always choose an $s$ and $r$ in these infinite sets proving that they're equal. But if I think about it like a person who might play with the sets first to see if the conjecture holds you will find it's always off by one. These two contradictory things I am trying to resolve and I can't split them enough in my head. Can anyone provide a better explanation?