0

In mathematics the word "strict" is often used, for example in "strictly convex", "strictly stronger", "strictly increasing".

For example, consider two statements:

  1. $f(x)=x^2+b$
  2. $f(x)=x^a+b$.

"1" is strictly stronger than "2". Can we say that "1" is strictly a special case of "2"?

I think this expression is natural but I find the terminology "strict" is never used with "special case".

High GPA
  • 3,776
  • 15
  • 44
  • 1
    $1$ is actually a special case of $2$. in that every function of type $1$ is also a function of type $2$. there are, of course, functions of type $2$ which are not functions of type $1$. – lulu Jul 26 '22 at 16:42
  • 1
    To your question, I think the more usual phrase would be "proper special case". That is, if every instance of $B$ is an instance of $A$ but some instances of $A$ are not instances of $B$, then $B$ is a proper special case of $A$. – lulu Jul 26 '22 at 16:44
  • @lulu Edited. You are right that 1 and 2 are inversed – High GPA Jul 26 '22 at 23:42
  • @lulu Re: "proper special case" I've never seen this phrase, neither. What did I miss? – High GPA Jul 26 '22 at 23:43
  • 1
    I think the "strict" or "proper" is usually implied by the phrase "special case", or at least it's usually obvious once the relationship has been pointed out. When/why would you want to emphasize the strictness? – Karl Jul 26 '22 at 23:54
  • @Karl Well if the math structure is getting complicated then it is not very obvious if one case is a proper special case of the other, or the two cases are somehow equivalent (i.e. both cases are the special cases of the other case). – High GPA Jul 27 '22 at 00:05
  • @lulu Well I think your terminology comes from "proper subset" – High GPA Jul 27 '22 at 00:10
  • 1
    @Karl I just checked Wikipedia that says that "proper" is implied by "special". However the wiki page does not cite sources. Is it a common sense in math or I should find a citation for this? – High GPA Jul 27 '22 at 00:14
  • I agree that "special" implies "proper" in common usage. If your argument somehow depends on this (which seems unusual), I'd state explicitly that the general thing is indeed more general. – Karl Jul 27 '22 at 04:13
  • I do not agree that special implies proper. In many cases it is obvious (or at least easy to demonstrate) that every $B$ is an $A$ but it is not at all clear whether or not there are $A$ which are not $B$. Even perfect numbers are a special case of perfect numbers....is it a proper special case? Should we not call it a special case until someone settles this issue? – lulu Jul 27 '22 at 06:25
  • @lulu Nice example; I've just expanded my reply to the OP's follow-up Question, to address it. – ryang Jul 27 '22 at 09:55

0 Answers0