Symbol for "not always equal"?

Question

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.

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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.

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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

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

Answer 1

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.

Answer 2

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.

Answer 3

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."

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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.

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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.

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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 $$ \lnot \forall a \forall b\;a = b. $$

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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.

Answer 4

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: $$\exists(x,y)\; f(x,y)= g(x,y) \quad\text{and}\quad\exists(x,y)\; f(x,y)\ne g(x,y).$$

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

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

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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).

Answer 5

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.

Answer 6

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".