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What symbol would I use if I wanted to express that, in the context of some binary relation $P$ implied from context, that $\exists (a,b)\in P: a\ne b$, but not to the extent that $\forall (a,b) \in P: a\ne b$.

The use of this would be if one were discussing a more restricted system, but then move to discussing a less restricted one. Like, "if we know for sure that $a\cdot b=b\cdot a$, then .... However, if $a\cdot b \mathrel{\rlap{=}\,?} b\cdot a$, then the previous reasoning doesn't apply, so ...". ("$\mathrel{\rlap{=}\,?}$" instead replaced with the real symbol)

AJMansfield
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  • Perhaps you mean $A \ne B$ – DanielV Nov 30 '13 at 00:23
  • I don't understand the question. The notation "$\exists a \in A\,\exists b \in B\, (a \ne b)$" does _not_ imply $\neg(\exists a \in A\,\exists b \in B\, (a = b))$. So what is wrong with this notation for what you are trying to express? – Trevor Wilson Nov 30 '13 at 00:24
  • @DanielV what I was after would be something that wouldn't necessarily require explicitly stating what sets $a$ and $b$ are in. Edited to make it clearer. – AJMansfield Dec 01 '13 at 18:01
  • @TrevorWilson I was looking to see if there was some kind of shorthand for the statement. – AJMansfield Dec 01 '13 at 18:02
  • I believe you are trying to introduce something like modal logic. I've never seen a use for it, but if this subject interests you then you should look it up. I believe they also have a (compound?) symbol for what you are trying to describe. – DanielV Dec 01 '13 at 18:28
  • I'm afraid I still don't understand. What does it mean to say $(a \ne b)$ after $a=b$ in the first paragraph? – Trevor Wilson Dec 01 '13 at 18:51
  • I'm pretty sure that the introduction of any symbol for "not necessarily equal" would defy the axioms of most non modal logics, btw, so finding a symbol for it outside of modal logic is probably impossible. – DanielV Dec 01 '13 at 19:22
  • @TrevorWilson I edited it again, does that help? – AJMansfield Dec 01 '13 at 19:37
  • The thing after _i.e._ is still not equivalent to the thing before, but I can ignore the thing after. What I am confused about is why you don't just say "there are distinct $a$ and $b$ such that $P(a,b)$"? Nothing about this says that there isn't also an $a$ such that $P(a,a)$, so I don't understand what the "but not to the extent that..." part is about. – Trevor Wilson Dec 02 '13 at 05:55
  • ...or did you perhaps not really mean it when you wrote "$\exists a,b \mathbin{\colon} P(a,b) \wedge a \ne b$"? If you had just written "$\exists a \,\exists b\, P(a,b)$" then this would not imply that there are _distinct_ $a$ and $b$ such that $P(a,b)$ holds. Perhaps this is what you are after? – Trevor Wilson Dec 02 '13 at 06:16
  • @TrevorWilson What I meant by $\exists a,b$ is, for domains $a\in A,b\in B$, that $\exists (a,b)\in A\times B$, not that $\exists a\in A,b\in B: a\ne b$. Thanks, I didn't realize that the notation I had used meant that $a,b$ are distinct. – AJMansfield Dec 02 '13 at 15:07
  • Your notation in the first paragraph of your question only means that $a$ and $b$ are distinct because it says so explicitly: $a \ne b$. Also, what do the sets $A$ and $B$ in your last comment have to do with the binary relation $P$ in the question? Is $P = A \times B$, or maybe just $P \subset A \times B$? – Trevor Wilson Dec 02 '13 at 15:55
  • @TrevorWilson You are right, sorry I just read your comment over again. What I want is a way to say "for the next part we are no longer constraining $a$ and $b$ to be equal". – AJMansfield Dec 05 '13 at 21:22
  • Your example should read "However if $a\cdot b\ne b\cdot a$" and nothing else, because you explicitly treated the case $a\cdot b=b\cdot a$ before. I guess that you should use another example. Also note that $a\text{ not necessarily equal to }b$ is not very different from $\text{any }a,b$. –  Sep 07 '16 at 09:09

5 Answers5

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tl;dr: the formal notation for this is:$~~~~\neg\square(a=b)$

Explanation:

Modal logic formally defines the following dual operators:

  • Operator "$\square$" meaning "it is necessary", and
  • Operator "$\lozenge$" meaning "it is possible".

For any proposition P, the following are true:

  • $\square P \leftrightarrow \neg \lozenge \neg P~~~~~~~~$, i.e. : "P is necessarily true" is equivalent to "P cannot possibly be false"
  • $\lozenge P \leftrightarrow \neg \square \neg P~~~~~~~~$, i.e. : "P may be true" is equivalent to saying "P is not necessarily false"

Therefore, if you're happy to concede that your 'equality' is a logical statement, then you can express such statements formally as follows: $$ \lozenge(A = B) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{(i.e. $A$ can be a $B$)}$$ or $$ \lozenge(A \neq B) ~~~~~~~~~\text{(i.e. $A$ can be something other than $B$)}$$ depending where you want to place the emphasis.
Or if you really want to express it in terms of necessity: $$ \neg\square(a = b) ~~~~~~~~~\text{(i.e. it is not necessary that a = b)}$$ etc.


PS. I suppose, if you preferred a "one-symbol-only" binary operator, like your $\overset?=$, you could define in your article the symbols $\overset{\square}=$, $\overset{\lozenge}=$, $\overset{\square}\neq$, and $\overset{\lozenge}\neq$ respectively in terms of the modal operator syntax stated above, and I'm sure these would be straightforward to follow in your text.

Having said that, if a strict logical statement is not needed in context, my preferred alternative answer here is the one given below by Dragon (i.e. \not\equiv: $\not\equiv$ ); to me this is fairly straightforward and intuitive, without requiring further explanation: stating that two quantities are not equivalent implies that they are independent variables that could nonetheless simply happen to take on an equal value.

Tasos Papastylianou
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There is a lecture series on Digital Signal Processing available on Youtube, in which a symbol appears which quite elegantly states "not necessarily equal to" by subscripting the "not equals" sign with the letter n.

Symbol for "not necessarily equal to"

Screenshot source: Prof. S.C Dutta Roy, Department of Electrical Engineering, IIT Delhi

rudolfbyker
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You may find $\equiv$, used to denote $\forall x (f(x)=g(x))$, to be useful, eg.:

$$a\cdot b \not\equiv b\cdot a$$

For reference, see Identity (mathematics) on Wikipedia.

Dragon
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It might be easiest to write out "however, if $a$ does not necessarily equal $b$" or "however, if $a$ doesn't have to equal $b$" ... upon a quick Google search, there doesn't seem to be a clear symbol for what you need, and it doesn't take that long to write it out.

Given that you won't be using this phrase as often as you would "there exists" or "for all" or "if and only if", it seems unnecessary to have a separate symbol for it.

Zubin Mukerjee
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I think this is a good point, as this kind of sentence is quite common in mathematics. However, one such symbol does not exist yet. So you should propose a new symbol for it, with a bit of creativity. Your equal sign with a question mark on it is not too bad, but I'm sure you can do better!

Abramo
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