What is the difference between these mathematical symbols

Question

1. $=$
2. $\cong$
3. $\equiv$

Can anyone explain me "clearly" what's the difference between these $3$ symbols. The more I google about them, the more I get confused.

Also, please tell me, how should I know which one to use? What are the hints?

Thank you :)

Answer 1

The first one is equality. The two objects on either side are (essentially) different names for the same thing.

The precise meaning of each of the other two depends on the context, so google can't know.

Often $\cong$ means "congruent" in geometry - hence its mathjax/LaTeX name "\cong".

The $\equiv$ is used for "congruent" in number theory. In other contexts it's read as "equivalent" (mathjax/LaTeX "\equiv") but what that means depends on the context.

When you are writing mathematics, use these symbols the same way the problem you're addressing uses them.

Answer 2

$=$ is the equals sign. You use it between two things which are the same. Not similar, not essentially the same. Exactly the same. (With a few exceptions, like in complexity theory with big-O notation.)

$\cong$ is usually used between things which are essentially the same. Isomorphic groups, homeomorphic topical spaces, congruent geometric objects, and so on.

$\equiv$ is used specifically in modular arithmetic, between two things which aren't necessarily equal, but considered to be the same in that specific context.

$\equiv$ also some times has a role where it is more similar to $=$, although not everyone uses it like this. When you see an expression like $3x+1=2x+3$, that's an equation. We're not saying that the two expressions on either side are the same, but only that $x$ has a value which makes the value on both sides the same. On the other hand, some people would choose to write $(x+1)^2\equiv x^2+2x+1$, because it's not about being equal for some values of $x$; the two expressions themselves are equal (we call this an identity). In a similar manner, $\equiv$ is some times used to denote a definition.

Answer 3

As with all symbols, their meaning can depend on the setting. However here are some common ways they're used:

1) "$=$" usually denotes equalities.

2) "$\cong$" is often used for isomorphisms.

3) For "$\equiv$" I'm aware of two common usages: In algebra it means congruence and in analysis it's an informal way of saying that two functions are the same at every point: Suppose $f,g\colon \mathbb{R}\rightarrow \mathbb{R}$ are two functions, then for me the following two expressions are synonymous:

• $f\equiv g$
• $\forall t \in \mathbb{R}:$ $f(t) = g(t)$.

Answer 4

The first one is equality in the intuitive sense that we all have e.g. 1+1=2.

The second one is used as "isomorph" in Algebra. There is an isomorphism between the two objects, meaning alltho there are some differences between them, they have the same structure. An example is $ℤ_2 ≅ _2$.

The last symbol stands for "congruent", which describe that two objects (often numbers) might be inequal, but are considered equal in some field. For instance: $10≡ 3\pmod7$