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Suppose that I am subtracting the size of two disjoint sets: for instance: $A$ and $B$ as $|A|-|B|.$ The result is presumably an integer. Now, my question is as follows:

When I think of subtraction, I think of removing some objects from a given amount of objects. In this case, how is subtraction best viewed? What are we ``removing'' from set A if there are no common elements with set B?

Kwame Brown
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  • Remove nothing is same as remove elements from empty set. – zkutch Aug 07 '21 at 15:20
  • When only considering the sizes of sets $A$ and $B$, the actual identity of the elements of each set is ignored, so "$A$ has no common elements with set $B$" isn't a relevant notion when thinking about $|A|-|B|$. I guess you could think of something like "for each element in $B$, remove an arbitrary element of $A$" for the case $|A| \ge |B|$, but I'm not quite sure what your motivation is for getting intuition for $|A|-|B|$. – angryavian Aug 07 '21 at 15:28
  • @angryavian Just notionally- for example, imagine the change in the number of people in a theatre. It is modelled as new coming in-old leaving. Now, in this case, both sets are disjoint- but we are still subtracting one set from the other. My question is this: is there a meaningful difference when considering subtraction as removing objects (as is usually taught in elementary school) vs considering the difference between two numbers? – Kwame Brown Aug 07 '21 at 15:31

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The assumption is that both $A$ and $B$ are finite.

In this context, I would think of it as follows:

a) Imagine all the elements of sets $A$ and $B$ each lined up in a long line, one line under the other.

b) Hence they are arranged in some order, it does not matter which.

c) Pair the elements off, one by one, and remove those pairs from the lines of elements.

d) Eventually you will reach the end of the set with fewer elements, and there will be none left.

e) The difference in set sizes is equal to the number of elements remaining in the larger set which have not been removed.

Prime Mover
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  • I see. I was wondering (and not sure if this related to the same question): but how is subtraction different when we are "taking away" a number from the other vs. when we are comparing two numbers? Is there even a difference? – Kwame Brown Aug 07 '21 at 15:44
  • @KwameBrown Suggest you study foundations of arithmetic where numbers are defined as sets, for example, the Von Neumann construction. – Prime Mover Aug 07 '21 at 15:49