Is |L1. L2| = |L1| x |L2|?

Question

The answer given is NO. 

I have a doubt. When concatenating two languages, we take the strings of the first language and concatenate it with each string of the second language. So if I go by the definition only, the above statement is correct.

But by taking a few examples, it seems that when we concatenate two languages, some of the strings comes out to be the same. The final concatenated language will contain strings less than |L1| * |L2|.

Therefore, the correct relation should have been |L1. L2| <= |L1| x |L2|.

Is this the right logic to answer this question? Or did I miss out on any conceptual reasons?

Answer 1

Yes that's correct. For eg. for ∑ as {0,1}

take L1 = {0,01} and L2 = {∈,1}

then L1.L2= {0,01,011}

so |L1.L2| = 3 and |L1|*|L2| = 2*2 =4

Comments

What's your opinion when Both the L1 & L2 are infinite Language ?
Then Still the Relation hold True ?