# PostgreSQL, 186 bytes#

    WITH y AS(SELECT * FROM t,LATERAL unnest(c)WITH ORDINALITY s(z,r))
    SELECT y.c,EVERY(u.z IS NULL)
    FROM y LEFT JOIN y u ON y.i=u.i AND y.r<>u.r AND y.z LIKE u.z||'%' GROUP BY y.c

Output:

[![enter image description here][1]][1]


No live demo this time. http://sqlfiddle.com supports only 9.3 and to run this demo 9.4 is required.

How it works:

1. Split string array with number and name it `y`
2. Get all y
3. `LEFT OUTER JOIN` to the same derived table based on the same `i`(id), but with different `oridinal` that start with prefix `y.z LIKE u.z||'%'`
4. Group result based on `c` (initial array) and use `EVERY` grouping function. If every row from second table `IS NULL` it means there is no prefixes.


<hr>

Input if someone is interested:

    CREATE TABLE t(i SERIAL,c text[]);
    
    INSERT INTO t(c)
    SELECT '{"Hello", "World"}'::text[]
    UNION ALL SELECT  '{"Code", "Golf", "Is", "Cool"}'
    UNION ALL SELECT  '{"1", "2", "3", "4", "5"}'
    UNION ALL SELECT  '{"This", "test", "case", "is", "true"}'         
    UNION ALL SELECT  '{"111", "010", "000", "1101", "1010", "1000", "0111", "0010", "1011","0110", "11001", "00110", "10011", "11000", "00111", "10010"}'
    UNION ALL SELECT  '{"4", "42"}'
    UNION ALL SELECT  '{"1", "2", "3", "34"}'                   
    UNION ALL SELECT  '{"This", "test", "case", "is", "false", "t"}'
    UNION ALL SELECT  '{"He", "said", "Hello"}'
    UNION ALL SELECT  '{"0", "00", "00001"}'
    UNION ALL SELECT  '{"Duplicate", "Duplicate", "Keys", "Keys"}';


  [1]: http://i.stack.imgur.com/Jaevb.png