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Updated post with better context:

I can do what I need to do using words. I am just trying to understand if there's notation I could use instead that would communicate this concept conclusively. The metric should be: There should be no doubt about what is being said. Words meet that metric perfectly.

You have two angle of attack sensors for a navigation system. Their readings go into $a$ and $b$. Because of errors, failures and other real-world factors, the values in $a$ and $b$ are not always equal. To smooth things out, the sensor data is passed through a set of convolution filters, error detection, failure mitigation and voting logic.

I am looking for notation that, as I said, symbolically communicates the very start of this process:

$$ Given \space that: \space a \space is \space not \space always \space equal \space to \space b \implies ... $$

The set of numbers $a$ and $b$ are in does not matter (assume $\mathbb{R}$ if it helps). The idea is to replace the words "not always equal" with one or more symbols.

This is what I have so far:

$$a \space\neg\forall= b$$

I read it as "not" "for all" "equal". That's the closest I think I have gotten to something symbolic that might be read as "a is not always equal to b". Not sure.

The use of $\not\equiv$ has been suggested. I can't see how this would ever be interpreted as "sometimes not equal" or, for that matter "sometimes equal" (which are equivalent characterizations). I don't think the distinction between equivalence and "sometimes equal/not-equal" is subtle at all.

Original post.

This is what most comments and answers referred to. It seems to have been confusing to some. Left here for context.

$$ \pm 2 \neq \pm 2 $$

This doesn't seem to cover it. Half the time $a$ and $b$ are equal. Is there a way to say "not always"? Or, alternatively, is the fact that $a$ and $b$ are not always equal enough to make $\neq$ tell the story accurately?

OK, let me see if I can concoct an example:

In proving that a function is injective you might start here:

$$ \forall a \forall b(f(a) = f(b) \implies a = b) $$

As you substitute the simplify the equation for, say $x^2 + 1$ you eventually end-up with $\pm a = \pm b$, which, of course, isn't always true. I am trying to understand how to say these are not always equal, therefore the function isn't injective.

Hope that helps.

EDIT:

The closest I think I've gotten after talking to someone IRL is something like:

$+a = +b$

$-a = -b$

$+a \neq -b$

$-a \neq +b$

And yet I still think one has to say "and therefore,..." to convey what needs to be communicated.

Or followed by

$$ \implies \forall a \forall b, a \neq b $$

martin's
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    This is very vague. Are these random variables? Something else? Context would help. – lulu Jun 14 '23 at 17:46
  • I don't think there is (generally accepted) general notation for what you are trying to say. The expression in the question is probably equivalent to $|a| \ne |b|$ but as @lulu notes, the question is ambiguous. – Ethan Bolker Jun 14 '23 at 17:48
  • Without any context at all, I'd tend to read it as $a \ne \pm b$, or $|a| \ne |b|$, either of which excludes any equality case. But it's an ambiguous notation which I haven't seen used, and I'd advise against using. – dxiv Jun 14 '23 at 17:48
  • What purpose would such notation serve you? – Derek H. Jun 14 '23 at 17:56
  • I think stating the equality and then specifying a set wherein it stands true , is what is usually used. Never seen such notation anywhere – Substitute_Y Jun 14 '23 at 18:30
  • Maybe my question isn't clear? How do you say two items are not necessarily equal? Or fully, 100% equal in all cased. I changed my question slightly because people seem to have gotten stuck with "a" and "b". – martin's Jun 14 '23 at 21:12
  • I guess something like $P(a=b)\ne1$? It is difficult to say since $a$ and $b$ are implied to differ in value "sometimes". You can define them as random variables of some distribution. – Scene Jun 14 '23 at 21:18
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    I've actually ran into this exact same problem before, and wasn't able to come up with a good notation to express it. I think it is best to just say "$a$ and $b$ are not necessarily equal" or, "unequal in general" or "not always equal", or some variation on that theme. – K.defaoite Jun 14 '23 at 21:35
  • When you work with ambiguous notation like $\pm 2$, how are you going to avoid ambiguity in more complex notation which contains the symbols $\pm 2$? What do you think the symbols $\pm 2$ mean anyway? Clearly you do not intend that $\pm 2$ is a shorthand notation for the set ${-2,+2}$, because from a set theory perspective it is unambiguously true that ${-2,+2}={-2,+2}$. – Lee Mosher Jun 14 '23 at 21:54
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    I would suggest you simply say what you want to say in a properly formatted sentence with a mix of natural language and mathematical notation, and not try to cram it into more compact notation. So, something like this: There exist $x,y \in {-2,+2}$ such that $x \ne y$. – Lee Mosher Jun 14 '23 at 21:55
  • Echoing the previous comment, when in doubt use plain words. This is commonly done to eliminate ambiguities elsewhere, for example "$a \pm b \pm c$ where the signs correspond" is understood to take two possible values, while "$a \pm b \pm c$ where the signs do not necessarily correspond" can take four. – dxiv Jun 14 '23 at 22:11
  • A quick note regarding the idea of just using words. Using symbols to communicate concepts is universal and independent of spoken language. Perhaps I betray my ten years of experience programming in APL. When possible, I prefer to use notation as a tool for thought and communication. – martin's Jun 15 '23 at 11:10
  • If "$\pm2$" makes any sense , then "$\pm2 \ne \pm2$" makes no sense , since every meaningful object is identical to itself. Misunderstandings are however unlikely in the cases , such a notation is used. It simply means that both signs are possible. – Peter Jun 15 '23 at 11:15
  • Please comment on my last edit. Thanks. – martin's Jun 15 '23 at 11:29
  • @martin's Your suggestion in your 'EDIT' is incomprehensible, whereas my answer's bullet point #3 conveys what you want to say. If your goal is effective communication and your question is about standard notation, then why not just choose from the 5 bulleted suggestions below? – ryang Jun 15 '23 at 12:32
  • Pipe down. This is a conversation. Not sure I agree with "not" "all" "equal" being incomprehensible. Still, I asked for feedback. With regards to your bullet list. I am not convinced that $\not\equiv$ would be interpreted as "not always equal". To me it means "never equivalent". I am looking for a notation that says that two variables are not always equal. I am not looking for a way to say that there exist a value that makes a function produce different numbers, or the same, or any variant of that. This is purely about how to say "a is not always equal to b" symbolically. Nothing more. – martin's Jun 15 '23 at 17:01
  • Feedback from others would be appreciated. Would you interpret $a \not\equiv b$ to mean "a is not always equal to b"? This would mean that sometimes they are equal and other times they are not. I would also appreciate comments on @ryang's proposed answer. Remember, this is about symbolically saying that a and b are not always equal. It has nothing to do with functions. – martin's Jun 15 '23 at 17:05
  • @martin's When your query "Does this make sense to anyone reading it?" is greeted in a mathematical conversation with "it's incomprehensible", I don't think "Pipe down" is a cohesive reply. $\quad$ "a is not always equal to b" means that sometimes they are equal and* other times they are not.* No: the claim "7 isn't always equal to 8" is, strictly speaking, true. – ryang Jun 15 '23 at 17:24
  • @martin's To me "$\not≡$" means "never equivalent". The symbol ≡ has multiple meanings; on this page, it symbolises a mathematical identity, not an equivalence relation. – ryang Jun 15 '23 at 20:32
  • @martin's In case you missed the previous edit note ("Keeps the OP's clarity edit while restoring the previous version so that existing answers are not rendered confusing and incoherent to read"): while you should of course make clarity edits, please do not simultaneously deface away your original queries to render existing answers confusing and irrelevant. (StackExchange is primarily for future users, and catering to the OP is just a secondary goal. This is also why Questions that have received Answers are not deletable by the OP.) Thank you! – ryang Jun 16 '23 at 11:13
  • The fact that this comment thread is so long is strong evidence that the last edit, suggesting that words will will work best in the OP's context, is the right answer to the question that ought to have been asked. – Ethan Bolker Jun 16 '23 at 11:21
  • @martin's: While personal preference and curiosity are all fine, let me nonetheless suggest that mathematics is not programming. When you write mathematics, you are communicating to other human beings, not to a computer. – Lee Mosher Jun 17 '23 at 04:34
  • @LeeMosher "you are communicating to other human beings, not to a computer. " That is precisely the reason for which I did not accept any of the proposed answers and concluded that words, for now, seem to be the only path. I put these answers in front of people well-schooled in mathematics and asked them to explain what they meant. Nobody --not one person-- came up with "not always equal". – martin's Jun 18 '23 at 06:22
  • Let me ask a question, although it requires some setup. The phrase "not always equal" does not make a complete sentence by itself. That phrase seems to indicate a relation between two objects, in which case the general format of a complete sentence which uses that phrase would be "$X$ is not always equal to $Y$". But, so far, $X$ and $Y$ are just symbols. We would like to replace those symbols by some kind of mathematical object in order to form a complete mathematical sentence. Now comes the question: What kind of mathematical objects may be substituted for $X$ and $Y$ in that phrase? – Lee Mosher Jun 18 '23 at 13:21
  • The reason I ask this question is that mathematical definitions and notations require a precise format, even when the definition is formulated in natural language. One would not define, or denote, the phrase "not always equal to" in isolation. One would, instead, define or denote a complete sentence of the form "$X$ is not always equal to $Y$". However, a mathematical definition also requires that one state what kinds of objects $X$ and $Y$ are allowed to be. – Lee Mosher Jun 18 '23 at 16:02
  • As an example, the phrase "is less than" is not defined in isolation, instead we define the sentence "$X$ is less than $Y$". But we also (usually) require that $X$ and $Y$ be real numbers. For a nonsensical example, we would not even bother defining the sentence "the set of real numbers is less than the permutation group $S_3$". – Lee Mosher Jun 18 '23 at 16:04
  • I understand. The way I look at it is that I am talking about notation that can be applied universally when it makes sense. $X$ and $Y$ can be many things. We could say two real numbers are not always equal, just as we say two numbers have a $<$ or $>$ relationship. Substitute sets for $X$ and $Y$ and the same things can be said with those symbols. I gave the example of proving a function is injective, where one starts with the assumption that $a=b$ to then either confirm the assumption or discover that "a is not always equal to b$. – martin's Jun 19 '23 at 14:27
  • Of course, symbols do have a relationship to what they are applied to, a context. If one were to make the categorical statement that $1 = 2$, everyone would understand this to be wrong. If, on the other hand, this came up at the end of a proof, it be used to close the proof with "and therefore...", which would make it a perfectly valid and proper equation (if that's the proper term). What does not change is that all readers would read $=$ as "a equals b$. – martin's Jun 19 '23 at 14:31
  • I think this is an interesting question. Of course, this is presumptuous of me because I am not a mathematician. I would think there would be value in being able to say $$\therefore\ \because X\ is\ not\ always\ equal\ to\ Y \Rightarrow$$ – martin's Jun 19 '23 at 14:40

6 Answers6

5

Given two entities, how do you symbolically say they are not always equal?

  1. These assertions are equivalent to one another:

    • the expressions $f(x,y)$ and $g(x,y)$ are not identically equal
    • $$ f(x,y)\not\equiv g(x,y) $$
    • $\neg\forall(x,y)\quad f(x,y)= g(x,y) $
    • $\exists(x,y)\quad f(x,y)\ne g(x,y)$
    • the equation $f(x,y)= g(x,y)$ is either conditional or inconsistent.

  2. On the other hand, the inequation $$\color\red{f(x,y)\ne g(x,y)},$$ in the absence of context, is ambiguous, and quite likely means that the left and right sides are never equal, i.e., $\forall(x,y)\;f(x,y)\ne g(x,y),$ rather than that they are not identically equal.

  3. Finally, to symbolically say that $f(x,y)$ and $g(x,y)$ are sometimes equal and sometimes unequal: $$\exists(x,y)\; f(x,y)= g(x,y) \quad\text{and}\quad\exists(x,y)\; f(x,y)\ne g(x,y).$$

$$ \pm 2 \neq \pm 2 $$ doesn't seem to cover it: half the time the left and right sides are equal.

  1. The string of symbols $$\color\red{\pm y\ne \pm x}$$ is ambiguous because it contains multiple ± signs: does it mean that $\;y\ne x\;$ or that $\;y\not\in\{-x,\,x\}$ ? More generally, does $$\color\red{z=\pm x\pm y}$$ mean

    • $z\in\{x+y,-x-y\}$

    or

    • $z\in\{x+y,\,x-y,\,-x+y,\,-x-y\} ?$

  2. While I have assumed that $$\color\red{z\ne\pm x}$$ means

    • $z\not\in\{-x,\,x\},$ that is, $z$ equals neither $-x$ nor $x,$

    conceivably, a potential (non-equivalent) alternative reading is

    • $z\ne -x \;\text{ or }\; z\ne x.$

Another example: in proving that the function $f,$ where $f(x)=x^2 + 1,$ is non-injective, you might start with $$ \forall a \forall b\;(f(a) = f(b) \implies a = b), \tag1$$ then eventually end up with $\pm a = \pm b.$ I am trying to understand how to say these are not always equal, therefore the function isn't injective.

Disproving statement $(1)$ requires exhibiting a specific counterexample rather than continuing to work with those universal quantifications, so there is no need to fiddle with plus-minus signs: just state that the counterexample $(a,b)=(-2,2)$ shows that there exists a case for which f(a)=f(b), but not a=b, is satisfied.

Needless to say, $$\forall(x,y)\quad f(x,y)\ne g(x,y)$$ ($f$ and $g$ are never equal) is a stronger assertion than $$f(x,y)\not\equiv g(x,y)$$($f$ and $g$ are not identically equal).

ryang
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  • My intent was to attempt to identify a universal symbolic way to say "not always equal". That's what the question is about, not injective proofs. That said, with injective proofs, one can't always pull numbers out of thin air. You have to take the reader to a point where they can understand the reason. In this case that the square root will result in a situation where it is well understood that opposite signs will make the function non-injective. I understand you only need one case. However, there are infinite cases, and just using +/-2 does not convey this. – martin's Jun 15 '23 at 11:37
  • @martin's My intent was to attempt to identify a universal symbolic way to say "not always equal". Did you not notice the answer, underneath its boldfaced verbal description, in the first section above?: $$\not\equiv.$$ – ryang Jun 15 '23 at 11:56
  • I did see that. I am not convinced it means "not always equal". This means a and b are sometimes equal and sometimes not equal. The negated triple bar does not translate to this. – martin's Jun 15 '23 at 12:16
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    @martin's $\not\equiv$ means a and b are sometimes equal and* sometimes not equal.* $\quad$ No it does not; it means precisely that its left and right sides are not always equal, in other words, for some tuple, they are unequal. For example, the assertions $$x\not\equiv x+1\A\not\equiv\pi r^2$$ are both correct, and this can be seen via $(x,r,A)=(1,1,3).$ – ryang Jun 15 '23 at 12:40
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This is somewhat of a joke, but I might write it as

$\pm 2\ \mathrm{mymn}\ \mp 2$

where "mymn" stands for "maybe yes, maybe no".

This phrase, btw, is taken from "The Further Adventures Of Nick Danger" in the Firesign Theater's record "How Can You Be In Two Places At Once When You're Not Anywhere At All". I highly recommend all of their works.

marty cohen
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Let's consider your example, a formula that says that a function $f$ is injective:

$$ \forall a \forall b(f(a) = f(b) \implies a = b). $$

Clearly, if we define $f$ by $f(x) = x + 1$ then $$ \forall a \forall b(f(a) = f(b) \implies a = b) $$ and therefore $f$ is injective.

But if instead we define $f$ by $f(x) = x^2 + 1,$ we find that $f(-2) = f(2)$ and therefore $$ \lnot \forall a \forall b(f(a) = f(b) \implies a = b). $$

In this expression, the two quantifiers $\forall a \forall b$ say "always," and by putting the negation operator $\lnot$ in front of them we change this to "not always."

On the other hand, suppose we use the notation $\not\forall{=}$ as proposed in the question, and apply it to this example:

$$ \forall a \forall b(f(a) = f(b) \implies a \mathrel{\not\forall{=}} b). $$

The $\forall a \forall b$ in this statement says that we can replace $a$ with any number we want, and then replace $b$ with any number we want, strip off the quantifiers, and have a true statement. For example,

$$ f(2) = f(2) \implies 2 \mathrel{\not\forall{=}} 2. $$

Really? The number $2$ is not always equal to itself?

On the other hand, $2\mathrel{\not\forall{=}}3$ might be considered true if "not always equal" includes the possibility that the two sides of the formula are never equal. If $\not\forall{=}$ means "sometimes equal, sometimes not equal" then $2\mathrel{\not\forall{=}}3$ is false.

What about the example $\pm2 \neq \pm2$? The problem here is the ambiguity of the $\pm$ symbol, which has already been explained in another answer. Out of context, we don't know what this formula means. You would have to embed it in a much more detailed example (such as the injective function example) in order to give it any meaning.

By itself, therefore, this isn't even an example.

Then we have the example where you find that $+a = +b$, $-a = -b$, $+a \neq -b$, and $-a \neq +b$.

In this case, the proposed conclusion, "therefore $\forall a \forall b\;a \neq b$" is false, because you have found cases in which the two sides actually are equal. In fact, if we take "$a = b$" literally, you have two equations that show this equation is always true and two inequalities that don't say anything about the equation $a = b$.

Again, the way to say that $a$ is not always equal to $b$ in an example like this is $$ \lnot \forall a \forall b\;a = b. $$

In summary, you have (as I write this) at least two answers that tell you how to say "not always equal" in formal notation and you have one joke answer.

You have made it clear (not in the question, but in comments) that what you're really asking is what you can use in place of the symbol $=$ in order to express this meaning. The answer to that, as far as I can see, is, "Don't do it!"

I have yet to see or conceive of any place where you can use such a symbol and still be writing useful mathematics. If you can come up with an actual useful example, I will reconsider this.

You have also expressed a strong desire not to "just use words" for this concept, because "Using symbols to communicate concepts is universal and independent of spoken language." Indeed there are many places in mathematics where symbols are the best way to express something; the legitimate mathematical use of the $=$ sign is an example. But you had to use words to ask your question, and not just because you lacked a symbol for "not always equal." I have over forty years of experience programming in various languages (including APL) and my takeaway from that experience is that while symbols are an excellent way to communicate with a machine, they are often a very poor way to communicate with human beings. If symbols were such a great universal language, for example, we wouldn't need comments in any of our code.

Sometimes there is a question whose answer is not what the questioner was looking for. This seems to me to be one of those questions. You can accept it, or you can refuse to accept it, but I don't think you'll do yourself any favors by refusing.

David K
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  • My intent was to attempt to identify a universal symbolic way to say "not always equal". That's what the question is about, not injective proofs. – martin's Jun 15 '23 at 11:31
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    @martin's I considered the injective proof because it was the only clear mathematical example you had given at the time. The nature of your question means that if we are to take it seriously, and not treat it as a joke, you must provide an example that shows why someone would want such a symbol. You have not done this, and I don't think you can do it. – David K Jun 15 '23 at 13:32
  • I truly do not understand why it seems to be so difficult to discuss such a simple concept. The idea that two things are not always equal and to use symbols to communicate that. Not too complicated. And yet, I get answers with things like $2 \neq 2$ is false. Well, how about a $\sqrt(4)$ is not always equal to $\sqrt(4)$. I don't think there's a rule that says one should take the same sign on both sides. You can't say they are always equal. You can't say they are always not equal. What do you say? How do you say it other than "not always equal". How do you do that symbolically? – martin's Jun 16 '23 at 11:46
  • @martin's By the usual definition, $\sqrt4=2$ so $\sqrt4=\sqrt4$ unambiguously. In short, there is a rule that you take the same sign on both sides. If, on the other hand, you mean "the solution" for $x$ in $x^2=4$, there are two distinct solutions which are never equal to each other. That seems simple enough to me. (By the way, https://aplwiki.com/wiki/Square_Root) – David K Jun 16 '23 at 12:14
  • If I assume the square root of a positive number is a positive number, I will crash very large objects that fly very, very high and very fast. I suspect you understand I meant $+/- \sqrt 4$ rather than just the principal root. BTW, I studied APL under Iverson back in the very early 80's (dating myself). Clearly there is answer to this question other than "use words". – martin's Jun 17 '23 at 01:36
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As I understand it $a$ and $b$ are time-dependent, so we should rather write $a(t)$ and $b(t)$ for their values at time $t$. A formal way to write that their values are not equal at every time is then simply $$\lnot ( \forall t : a(t) = b(t)).$$

If we want to refer to this property often in some formal context, and perhaps emphasize that they at least should be almost equal, we can of course adopt some notation like $$a \approx b.$$

But if it's in a non-formal context, especially if only a few times, just write "$a$ and $b$ are not always equal".

md2perpe
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  • I appreciate the perspective. Every example, even real-life scenarios, seems to confuse people into focusing on the details of that example rather than the idea of what my question was about. In fact, I don't think an example is needed. This was a search for a way to say "this is not always equal to that" using notation, not words. The minute we start introducing new variables the problem changes. – martin's Jun 18 '23 at 06:18
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If two comparable mathematical entities, say, a and b, are not necessarily equal, then either a < b, or a = b, or a > b. There are two, somewhat equivalent Unicode symbols that describe this relation between a and b:

a ⋚ b is read as: a is less than, equal to, or greater than b.

a ⋛ b is read as: a is greater than, equal to, or less than b.

I recently used the first symbol in a math seminar presentation when comparing the various syntax and semantic interpretations of MOD in mathematics and computer science. For example in number theory, a ≡ b MOD m means that a and b are members of the same equivalence class, which means that they could be equal but not necessarily so.

James D. Klein
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I attempted to find an answer here. I conducted extensive searches and even explored various ideas using GPT4. I also consulted a couple of friends from MIT and WPI. Bottom line, there does not seem to be a universally-accepted mathematical symbol (or combination) that precisely and unequivocally conveys the idea of "not always equal" or "not always not equal" (which mean the same thing).

Therefore, it seems a good idea to close this inquiry with the only conclusion I have been able to reach: Use words.

$$a \space is \space not \space always \space equal \space to \space b$$

That, ignoring having to understand English, communicates the idea precisely to anyone reading it.

martin's
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