Problem with symbology

Question

I know this may be a pretty basic question, but what is the difference between $\approx , =,\cong, \text{and}\sim $ ?

I had problem while changing schools and now I am confused.

Answer 1

$=$ means "equal". As in: $x=5$

$\approx$ means "approximately equal" as in: $\pi \approx 3.1415926535898$ .  It is also sometimes used to mean "isomorphic" in group theory.

$\cong$ means "congruent", typically geometrically congruent, as in: $\triangle ABC \cong \triangle DEF$ means the two triangles have the same angles and lengths.

$\sim$ means "similar".  Sometimes used to indicate only weak approximation.  In geometry it may indicate shapes are similar but not congruent (ie: have the same angles but not necessarily the same scale).  In set theory it is used to indicate elements belong to the same equivalent class.  In probability theory it can mean a random variable has a distribution: $X\sim \mathcal{U}(0,1)$ means $X$ has uniform distribution over the interval $[0, 1]$.

Eds: These and other symbols can be found on wikipedia.

Answer 2

That's a nice question, but it depends to much on the author. Understand, symbols are a matter of convention, they are not some kind of "universal rule" that every mathematician has to follow. Take Frege's Begriffsschrift for example, it uses some very weird symbolism, totally different from the one used in his time. I can also mention the symbolism used in Principia Mathematica, from Russell and Whitehead, which is just some adaptation from Peano's notation. Anyway, in conclusion, the meaning of this symbols vary from author to author, but it appears to be some general agreement in respect to $=$, which means just "equal". The other ones, as I said, are just restricted to the formal language used in the text.

P.S.: More serious authors define the basic symbols of their language right on the beginning of the text, in attempt to clarify the logical notions used in the paper. Symbols are also defined in some symbolic indexes. Have a nice day.