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I'm working on some problems involving set theory, more specifically subsets. I think I understand how it works, but just wanted to make sure that my thought process is correct.

For something to be a subset of something else all of the possible outputs of the set would need to be contained in set. For subsets to be equal they need have all of the exact same outputs, even if their inputs are different to get the outputs.

$A=\{2a:a∈\mathbb{Z}\}$

$ B=\{b:b∈\mathbb{Z}\}$

$A⊂B$

$A$ is a subset of $B$ because all possible outcomes of $A$ for any number plugged in equal a number that is in $B$

$A=\{a:a∈\mathbb{Z}\}$

$ B=\{b−1:b∈\mathbb{Z}\}$

These sets are equal and also contained within each other because any output in $A$ can be found in $B$ with a different input. Am I thinking about these correctly?

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  • $\begingroup$ Yes, your conclusions are correct, though your use of "input" and "output" is non-standard. What you call "outputs" are the elements of the sets. You think of a set description as an algorithm: you plug in the integers, and out come other integers. If that helps you understand what's going on, it's OK, but, as I said, the nomenclature is not commonly used. $\endgroup$ – Fabio Somenzi Jan 23 '18 at 1:53
  • $\begingroup$ It sounds like you have the wrong idea when you're talking about "outputs" of a set. A set doesn't do anything; it just sits there and has elements (or members). And it certainly doesn't have "inputs". Do you realize that when you write $A=\{A:a\in\mathbb Z\}$ you're just defining $A$ as a different name for the set already known as $\mathbb Z$? You could write $A=\mathbb Z$ instead. I suspect you may be confusing sets and functions. $\endgroup$ – Henning Makholm Jan 23 '18 at 1:53
  • $\begingroup$ Thanks for the help. I was thinking about the first part of the set (the 2a and a in the first example I gave) sort of like a function that gave all of the elements of the set. That's where I was coming up with inputs/outputs like a function. $\endgroup$ – prestigem Jan 23 '18 at 2:14
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Usually when we (at least, when I) think about sets, we think about members, rather than inputs and outputs. The notion of input and output makes sense when talking about functions; when talking about sets, either an object is in the set or it is not; the set doesn't do anything to it like a function does, so inputs and outputs don't really make sense.

In your example, the set $B=\{b-1 : b\in \mathbb{Z}\}$ is described by means of a function $b-1$, so I understand where you brought input and outputs in here. However, this is just a way this specific set was described; the function is there as a test to check if something is in the set or not. I could also describe a set as $\{3,5,9,764\}$, without referencing a function. So, in general, input and output is not the best conceptual framework for thinking about sets and subsets.

Thinking of a set as a collection of objects, a subset is simply a collection of some (or all) of those same objects.

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  • $\begingroup$ Thanks for the explanation. I was definitely thinking about it in terms of functions because I was seeing a function within the set $\endgroup$ – prestigem Jan 23 '18 at 2:16
  • $\begingroup$ @prestigem Yes, these sets are described by functions. And, if you translate the membership notion of subset into function language and apply it to the specific case of sets described by functions, you come out with a description like yours in the question. $\endgroup$ – Y. Forman Jan 23 '18 at 2:19
  • $\begingroup$ Quick follow up question. I'm looking at another set A = {n/2 | n ∈ N}. Does the fact that n must be a natural number mean that elements of the set cannot be real numbers? $\endgroup$ – prestigem Jan 23 '18 at 2:45
  • $\begingroup$ @prestigem In most (but not all) contexts we'd think of the natural numbers as a subset of the reals, so a natural number is a real number. Is this not appropriate to your context? $\endgroup$ – Y. Forman Jan 23 '18 at 2:51
  • $\begingroup$ Yes, I understand that natural numbers would be the subset of reals. I think I may have worded my question incorrectly. I'm trying to figure out if real numbers like 1/2 or 3/2 would be permitted to be contained within set A, even though n is a natural number. $\endgroup$ – prestigem Jan 23 '18 at 3:15
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Just think of them as collections and check whether one is a sub collection of the other . Terms like input and output are non standard .. those things are used for sequences in some non standard books .. Furthermore writing a set in set builder form is not always possible, therefore there may be nothing like input and output in those sets ..

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