What makes a conditional statement to be considered a theorem instead of a rule?

Question

The background

While reading a book on discrete mathematics (Discrete mathematics and its applications by Kenneth H. Rosen) on the topic of counting I found the following definitions:

Extracted definition of the "product rule" from the mentioned book

Extracted definition of the "sum rule" from the mentioned book

Extracted definition of the "pigeonhole principle" from the mentioned book

The question

Why did the author decided "the product rule" and "the sum rule" to be considered as "rule"s instead of "theorem"s just as happens with the "pigeonhole principle"? Both groups (1) the sum rule and the product rule, (2) and the pigeonhole principle are conditional statements. I can't distinguish a property that (2) has that (1) doesn't have.

I am asking this because the way I take notes is by clasifying what I found interesting in the following groups: definition, theorems, notations. Because of the fact that the author didn't consider "the product rule" and "the sum rule" as theorems, I just was wondering if it was correct to include them in the group of theorems from my notes.

You may check this one https://math.stackexchange.com/questions/24758/difference-between-a-theorem-and-a-law — Alapan Das, May 14 '20 at 05:07
I suspect a rule is a way of doing something, and a theorem is a proven fact. Of course a rule often comes from a theorem, but it can be from a definition eg the rule for adding matrices. — Peter, May 14 '20 at 05:13
They are definitely more theorems then they are definitions or notations. — Greg Martin, May 14 '20 at 06:09
Because they express a procedure: a "recipe" to compute a result. — Mauro ALLEGRANZA, May 14 '20 at 06:14

What makes a conditional statement to be considered a theorem instead of a rule?

The background

The question

0 Answers0