An issue with the difference between natural numbers when they represent the number of elements of sets

Question

For example, we have a PDF document with 13 chapters and we want only to print chapters 6 to 13. Then, one could think that the number of chapters that would be printed is 13 - 6 = 7, but the real number is 8.

I see that 6 and 13 would represent the number of elements between the sets $\{1,2,...,6\}$ and $\{1,2,...,13\}$, respectively, and that $\{1,2,...,6\} - \{1,2,...,6\}=\{6,7,..., 13\}$. Where is the problem? How could be it explained easily to kids so they know how to do with problems like the example above?

Answer 1

In English: When the range is from m to n inclusive, m must of course be part of the range. The problem is, when you take 1..m away from 1..n, you are taking m out of the range. You have to add it back in. Hence the size of the range m..n is NOT (m - n), it is (m - n + 1).

Answer 2

$$\{1,2,\ldots,6\} - \{1,2,\ldots,6\}=\emptyset\neq\{6,7,\ldots, 13\}$$ $$\{1,2,\ldots,13\} - \{1,2,\ldots,6\}=\{7,..., 13\}\neq\{6,7,..., 13\}$$

subtraction doesn't include the 6 in the result. if you do, you get the equivalent of: $$\{1,2,\ldots,13\} - \{1,2,\ldots,5\}=\{6,..., 13\}$$

Answer 3

In your example, I would think of Chapter 5 as the "zero" of this operation. Though then again, I doubt any kid, other than perhaps young Sheldon, would care at all about chapters in an Adobe PDF document.

I would use numbered balls or numbered tiles. Then the kids can count for themselves and see if the subset has as many elements as they thought it would.