Note that bash does not natively support a bool type operator. All the commands/built-ins return an exit code to the shell depending upon its success/failure status. A code 0 to the shell means the operation is success and any non-zero would represent a failure.
So in bash, you do the comparison and need to set the bool strings explicitly, something like
[[ $a == $b ]] && echo true || echo false
Note that using echo true should not confused with the built-ins /bin/true and /bin/false which explicitly set the exit code to the shell to 0 and 1 respectively. The echo statement just prints out the string mentioned to the standard output.
Also note that [[ is a shell keyword extension to the POSIX-ly available [ construct. The former is a added to the bourne again shell and may not available in other POSIX compliant shells. If you are looking for a portable way of doing this, use the [ with the case construct as
[ "$a" -eq "$b" ]
case $? in
    0) printf '%s\n' "true" ;;
    *) printf '%s\n' "false" ;;
esac