Glue-concatenation v.s concatenation
The statement does not hold.

For $A = \{a, aaba\}$ and $B = \{a, abaa\}$ we find

$|g(A,B) = \{a, abaa, aaba, aababaa\}| = 4 &gt; 3 = |\{aa, aabaa, aabaabaa\}|  = |A\cdot B|$.

As a sidenote, I'd say this could actually be a worthwile exercise task, once students are reasonably experienced with standard proof techniques. After all, you only really learn how to attack such problems from practising.