What is the difference between the following commands and why do they not produce the same output?:
echo $[ "a" == "a" ]
1
echo $(( "a" == "a" ))
1
[ "a" == "a" ]; echo $?;
0
echo $[ "a" == "a" ]
uses the deprecated integer expansion syntax, which has been replaced by $(( ... )).
echo $(( "a" == "a" ))
uses what is called arithmetic expansion: inside $(( ... )), you can have logical and numerical expressions that get calculated before returning the value.
[ "a" == "a" ]; echo $?
does a logical test in standard bash fashion: the return code of a program is 0 when that program executes successfully, and non-zero otherwise. [ is actually another name for the program called test, which returns success (i.e., 0) when the expression it tests is true.
Not in your list, but related:
(( "a" == "a" )); echo $?
0
this does arithmetic expansion, and returns success (0) if the result is true, rather than producing the value of the expression itself.
[[ "a" == "a" ]]; echo $?
0
[[ ... ]] is bash syntax for doing conditional tests directly instead of calling the program test.
Some sample usages
You would use $(( ... )) to initialize variables or echo output, as you did:
foo=$(( 1 + 4))
echo $(( 5 + 7 ))
You would use [ ... ], (( ... )) and [[ ... ]] in conditional statements:
if [[ $a == bar ]]; then
echo $a is bar
fi
i=0
while (( i < 10 )); do
echo i=$i
i=$(( i + 1 ))
done