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?
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.