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I'm interested in knowing what is the meaning of the various equality symbols: $=,\sim, \cong,\approx,\equiv$.

For example, the speed of a car $V$ in m/s: what would be the meaning of each of these statements? $$V = 30\\ V\sim 30\\ V \cong 30\\ V \approx 30\\ V \equiv 30$$

m_power
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    None of those, except maybe "$=$", have a consistent context independent meaning. ~ and $\cong$ are usually used to say "approximately equal". – Gregory Grant May 06 '15 at 15:14
  • @GregoryGrant $=$ is the worst offender; it has the most possible contexts of them all. – Emily May 06 '15 at 21:13
  • @Arkamis Ok fair enough, I was just saying it always usually means "equal" in whatever context. While $\equiv$ sometimes means "congruent", sometimes "equivalent", sometimes "is defined as", etc... – Gregory Grant May 06 '15 at 22:16

4 Answers4

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Maybe instead of handling your example, because the context is not always relevant, let's look at possible groupings of the symbols.

Equality

  • $=$ is usually used for equality.
  • $\equiv$ is occasionally used for "identically equal to," which is in a sense stronger than equality, by denoting that the thing on the left and the thing on the right are equal in a sense that they are identities of each other. For instance, $f(x) = 0$ might be interpreted as "when $f(x)$ equals $0$," but $f(x) \equiv 0$ means "$f$ is zero everywhere." The existence of this usage is because of my next example.

Conditional Equality

  • $=$ is a horrible symbol. In algebra, we write things like $x^2+x+1 = 0$. What we mean when we write this is not that the quantity on the left is always the quantity on the right, but rather that it is conditionally so at some points, q.v. at the zeros of $x^2+x+1$.

Equivalence

  • $\sim$ is often used to denote a generic equivalence relation, e.g. "$x\sim y$ if $x-y\in\mathbb{Q}$."
  • $R$ is also often used for the same purposes.

Definition

  • $=$ is often used to define things. "Let $a=3$. Let $X = \{ x : \langle x,y\rangle = 0 \forall y\in M\}$." And so forth. It is clear from context, but the meaning of $=$ is different than the aforementioned cases. Also used in programming. int n = 5;
  • $\stackrel{\Delta}{=}$ is used as "define the thing on the left as the thing on the right." Often used in blackboard writing, as it is quick and easy.
  • $:=$ Used in programming at times; e.g. the Pascal language, Maple, and some others.
  • $\stackrel{\textrm{def}}{=}$ my personal favorite way to denote a definition. Clear and unambiguous and a pain to type in LaTeX. (That's what macros are for.)

Approximation

  • $\approx$ This is used often to say "the thing on the left is equalish to the thing on the right." Example: $\pi \approx 3.14$.
  • $\cong$ Used sometimes in engineering, this is horrible.
  • $\sim$ Used less frequently, but still comes up on occasion.

Distributed Like

  • $\sim$ is used in probability to declare that a random variable has a distribution of some sort, e.g. $X \sim \mathcal{N}(\mu,\sigma^2)$.
  • $\sim$ is also used in asymptotics and computational fields to describe the order of something, e.g. $e(x) \sim \mathcal{O}(h^4)$.

Shaped Like

  • $\cong$ is used to denote isomorphisms, e.g. $A_4 \cong PSL(2,3)$.
  • $\cong$ is used in geometry to denote when two shapes are congruent.

This list is by no means complete. However, you will see most of these uses if you even just browsed this site for a week or so.

Emily
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  • Regarding $x^2+x+1=0$ I regard the only problem as missing words which we never mention... (ASSUME) $x^2+x+1=0$ (is true) and from here we deduce the values of $x$ for which the equation is true. – JP McCarthy May 06 '15 at 21:15
  • @JpMcCarthy The problem, in my opinion, would be the use of the symbol $x$. $x$ in $y=x+2$ means $x: x\in Dom(f)$, while in $x+2=0$, it is being used as an element $x\in Dom(f) : x+2=0$. – YoTengoUnLCD May 06 '15 at 21:19
  • About the conditional equality example, in that context I've never felt that there's that issue because there are two possible ways to quantify the variable x: universal and existential. I just always take it to mean that x is existentially quantified in that sort of context, unless otherwise stated. I think of it more of a variable quantification issue than an issue with the equality symbol itself. – David May 06 '15 at 22:17
  • I wish I could upvote twice just for "Used sometimes in engineering, this is horrible." – Avi May 07 '15 at 15:00
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If your example is about speed, then this a question for physicists and not for logicians.

The most important thing is that these symbols are precisely that - symbols. They can be defined to mean whatever you want them to. Of course, we try to keep some common basis of definitions across a particular topic. But even the standard equal sign can be used for different things: I could equate terms (like $2+2=4$), sets, whatever.

Now, back to speed, and the different "equality" symbols:

  • $V=x$
    • $V$ is precisely equal to $x$;
  • $V\simeq x$ and/or $V\approx x$
    • $V$ is approximately equal to $x$;
  • $V\sim x$
    • $V$ is asymptotic to $x$;
  • $V\equiv x$
    • $V$ is defined as $x$ (physicists tend to use this symbol for definition, whereas mathematicians might mean "congruence", and logicians "equivalence").

My point is, you need to be aware of what topic (or branch of mathematics) you're working on, as these symbols (and many others) will have their own interpretation.

jvriesem
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Demosthene
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Many of the symbols you list, unless they have meanings I'm unaware of, are meaningless in this context. In general:

$=$ means, as I assume you know, is 'equal to' (there is a certain subtlety here, but I'm hoping you don't want to go into it).

$\approx$ means 'approximately equal to.'

$\sim$ means, in the contexts I'm aware of, 'is asymptotic to,' typically as the arguments go to infinity (although it can be any other value).

$\cong$ and $\equiv$ both mean 'are congruent to' (once again, in the contexts I know).

For a more comprehensive list and explanation, see this Wikipedia page.

Avi
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    Doesn't apply in this context, but $\sim$ in the context of probability theory often means "is distributed as". – A. Donda May 06 '15 at 15:33
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    @A.Donda And it can also be used in logic as a substitute for the negation symbol $\neg$, or as an equivalence relation, or in physics as a characterization of the order of magnitude. – Demosthene May 06 '15 at 15:44
  • I was thinking of asymptotic analysis and number theory. – Avi May 06 '15 at 15:51
  • From the link, ∼ could also mean "order of magnitude". – m_power May 06 '15 at 16:12
  • ∼ is used in linear algebra to denote a transformation to a matrix. – OFRBG May 06 '15 at 18:09
  • @m_power At least in asymptotic analysis, $\sim$ means not only on the same order but asymptotic to (that is, with constant $1$). – Avi May 06 '15 at 18:55
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    Basically one could say that ∼ means that there's a relationship between the left-hand and the right-hand terms, but it's not an exact relationship, more of a bound or range. But it's an ambiguous operator that can get very different meanings depending on the context and the field of study (such as the one pointed by @Demosthene, in logic, ~ can mean negation as failure, but we can then consider that it's a different operator since in logic it's unary, instead of binary as OP asked). – gaborous May 06 '15 at 21:03
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ISO 80000-2:2009 entitled "Quantities and units — Part 2: Mathematical signs and symbols to be used in the natural sciences and technology" should cover them.

null
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    That ISO standard is as relevant to usage of mathematical notation as to making chocolate cakes. – Mariano Suárez-Álvarez May 06 '15 at 21:15
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    @MarianoSuárez-Alvarez Hey now, there is not a field wherein ISO does not find a way to apply any standard. – Emily May 06 '15 at 21:16
  • ISO 20000 (food safety management) applies more directly to making chocolate cakes. I'm sorry that I couldn't find something more specific, but I'm sure that it's only a matter of time. – Toby Bartels Jan 22 '16 at 19:15