Double parenthesis with and without dollar

Is $(...) the same as (...) in Bash?

Also, is $((...)) the same as ((...))?

Also, is ${...} the same as {...}?

More generally what does the dollar sign stand for? Thank you.

• $(...) means execute the command in the parens in a subshell and return its stdout. Example:

    $ echo "The current date is $(date)"
    The current date is Mon Jul  6 14:27:59 PDT 2015
  

• (...) means run the commands listed in the parens in a subshell. Example:

    $ a=1; (a=2; echo "inside: a=$a"); echo "outside: a=$a"
    inside: a=2
    outside: a=1
  

• $((...)) means perform arithmetic and return the result of the calculation. Example:

    $ a=$((2+3)); echo "a=$a"
    a=5
  

• ((...)) means perform arithmetic, possibly changing the values of shell variables, but don't return its result. Example:

    $ ((a=2+3)); echo "a=$a"
    a=5
  

• ${...} means return the value of the shell variable named in the braces. Example:

    $ echo ${SHELL}
    /bin/bash
  

• {...} means execute the commands in the braces as a group. Example:

    $ false || { echo "We failed"; exit 1; }
    We failed
  

> More generally what does the dollar sign stand for?

It means whatever it means in the given context.

Adding to the answer above:

1. [..] is used in conditions or logical expressions. Example:

$ VAR=2
$ if [ $VAR -eq 2 ]

> then
> echo 'yes'
> fi
yes

2. [[...]] offers extended functionality to single square brackets. Particularly, it is useful for =~ operator (used in regular expressions). Example:

$ VAR='some string'
$ if [[ $VAR =~ [a-z] ]]; then
> echo 'is alphabetic'
> fi
is alphabetic

Reference: > https://linuxconfig.org/bash-scripting-parenthesis-explained