If I have a set A, comprising of numbers from 1 to 10: $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Let's say I want to make another set by "including" all even numbers:
$\{2, 4, 6, 8, 10\}$
Or I wanted to make a different set by "excluding" all odd numbers:
$\{2, 4, 6, 8, 10\}$
These sets are of course the same. So is it correct to say that inclusion/exclusion are synonymous when it comes to set theory, as they're just different ways of building a set?
This might sound trivial, but I have a reason for asking: I want to understand if inclusion and exclusion are "commutative" properties, i.e. it doesn't matter in which order you apply them.
For example, let's say I make an operation to "filter" my set, by including all even numbers as we did before, producing set B
$B = \{2, 4, 6, 8, 10\}$
And then a separate operation to "exclude" any numbers less than 6 from set B, resulting in set C:
$C = \{6, 8, 10\}$
What if I started with A and applied the operations the other way around? First remove all numbers less than 6:
$B = \{6, 7, 8, 9, 10\}$
Then filter B to "include" only even numbers:
$C = \{6, 8, 10\}$
It seems intuitively to me that the result will always be the same no matter which order you apply the operation. Is this true for all cases no matter the set, however? Is there a way to prove that applying "filters" to a set (I'm not sure of the proper term) will always be commutative?
So in summary, there are two questions here:
- Are the notions of "inclusion" and "exclusion" really synonymous from the point of view of applying an operation to a set to produce a subset?
- Will applying these operations to produce a subset of a set always be commutative, i.e. produce the same result?