The meaning of various equality symbols

Question

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? $$V = 30\\ V\sim 30\\ V \cong 30\\ V \approx 30\\ V \equiv 30$$

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.