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I have seen in some places; single relational symbols (particularly $\neq , \lt, \gt, \leqslant, \geqslant $) are being used for more than one sense; usually 2 sense, such as (1.) "exactly" or "always" and (2.) "Not necessarily", or "There may be exceptions". My question is; is there a more specific notation for these 2 usage (1.) "exactly" or "always" and (2.) "Not necessarily"? Maybe a set-theory notation?

Such as

$$ \begin{array}{lr} a-b \neq b-a ~~~&~~~ ...1 \end{array}$$

and also

$$ \begin{array}{lr} a \neq a+1 ~~~&~~~ ...2 \end{array}$$

But there are difference between the sense of these two $\neq$ s.

The relationship- 1. is an attempt to demonstrate subtraction is not commutative. But the $\neq$ sign is not used in the sense that $a-b$ will be always unequal with $b-a$. Such as if $a=0$ and $b=0$ then $(a-b) = 0-0 = (b-a) = 0-0$. For any given values of $a-b$ we can not "conclude" $a-b$ would be always unequal to $b-a$.

But the relationship 2. tells the terms $a$ and $a+1$ will be always unequal. And in return, for any given values for $a$ we can conclude $a+1$ is always unequal from $a$.

Now is there any notation to distinguish "Always Unequal" and "Occasionally unequal"? (and for similar situations occur for inequalities? )

NB. In case of Relationship-1, I could denote the exception in third brackets ( $[ ]$ ) at right hand side of the expression; but in cases all the possible exception could not be figured out; could there be any "General" notation that there are "some" exceptions?

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    $\begingroup$ You can qualify each statement as such: $\forall a~:~ a\neq a+1$. Meanwhile $\exists (a,b)~:~ a-b\neq b-a$ $\endgroup$ – JMoravitz Jun 27 '18 at 17:13
  • $\begingroup$ @JMoravitz Thanx Plz. Make it an answer. $\endgroup$ – Always Confused Jun 27 '18 at 17:20
  • $\begingroup$ Could anyone plz edit my question to put ...1 and ...2 at right hand side (with a few space distance) of respective relations? (Preferably using MathJax commands). $\endgroup$ – Always Confused Jun 27 '18 at 17:25
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    $\begingroup$ \begin{array}{lr} a\neq a+1~~~&~~~...2\end{array} yields $$\begin{array}{lr} a\neq a+1~~~&~~~...2\end{array}$$. See more at this page. $\endgroup$ – JMoravitz Jun 27 '18 at 17:27
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    $\begingroup$ Honestly, I would just use words. You can say "It is not always true that $a-b = b-a$." Usually in math writing we eschew using logical symbols like $\neg, \forall, \exists$ unless there is a special reason. $\endgroup$ – Jair Taylor Jun 27 '18 at 17:54
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Quantifiers are needed. No truth-value can be assigned to "$a-b=b-a$" as it stands.

The sentence $\forall a \forall b\;(a-b=b-a)$ is false.

The sentence $\exists a \exists b\;(a-b=b-a)$ is true.

A sentence such as "$a-b=b-a$ is true for some, but not all, $a,b$" is often expressed as "In general, $a-b\ne b-a.$"

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  • $\begingroup$ Thanks, Quantifier would be the concept I was looking for. $\endgroup$ – Always Confused Jun 27 '18 at 18:37

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