This is a precedence problem with your ternary statement. If you parenthesise:

($#arr == -1 ) ? ($string = 'True' ): ($string = 'False');
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then the output is as expected. However it is probably better written as:

$string = ($#arr == -1 ) ? 'True' : 'False';
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By the way, $#arr is not the length of the array, it is the highest index number. So your ternary could be written like this:

$string = @arr ? 'False' : 'True';
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The array in scalar context gives the number of elements (zero is false).