Another confusing error from someone else's Cygwin setup

[David Karr <davidmichaelkarr@gmail.com>]

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I'm seeing a problem with someone else's Cygwin setup, sort of similar to a
problem I asked about a couple of weeks ago, in that it's a problem with
the same user, but seemingly a completely different problem.

He is using a Bash script that I wrote, and he gets a seemingly nonsensical
error that I don't understand.

The script starts out pretty simply, just like this:
------------------
#! /bin/bash
#set -x
main() {
if [ "$1" == "" ]; then
usage;
exit;
fi
...
-----------------

He was getting a weird error on line 3, just saying this:
---------------------
...: line 3: syntax error near unexpected token `$'{\r''
...: line 3: `main() {
---------------

This was pretty perplexing, so I asked him to uncomment the "set -x" line
to see if that provided any useful information, and that fails with a
different error:
-------------------
: invalid option...: line 2: set: -
set: usage: set [-abefhkmnptuvxBCEHPT] [-o option-name] [--] [-] [arg ...]
------------------

This also makes no sense to me.  I compared our "uname -a" outputs, and
they are almost identical. However, I then had him run "bash --version",
and I compared it to mine.  Ironically, I'm using an OLDER version of bash
than he is. I'm on v4.4.12(3)-release, and he's on v5.2.15(3)-release.  Is
there something in 5.x versions of Bash that could cause these issues?