What is the language of this regular expression $0^*$ concatenated with $1^* (0^*1^*$).

Question

What is the language of this regular expression $0^*$ concatenated with $1^* (0^*1^*)$? $$ \begin{align} L(0^*1^*) &= L(0^*)L(1^*)\\ &= {λ, 0, 00, ...}{~λ, 1, 11, ...}\\ &= {λ, 0, 1, 01, 011, 001, 0011, 0001,...}\\ \end{align} $$ This is what I have. I Don't know if it is correct. Can you explain how the concatenation work for regular languages? Thanks.

Answer 1

Concatenation simply means writing one string after the other.

The concatentation of two sets is the set containing all possible concatenations of one string from the first set followed by one string from the second set. It does not matter whether the sets/languages are regular or not.

In your case: $$ \begin{align} L(0^*\cdot 1^*) &= L(0^*)\cdot L(1^*)\\ &= \{λ, 0, 00, ...\}\cdot\{~λ, 1, 11, ...\}\\ &= \{λ (=\lambda\cdot\lambda), 0(=0\cdot\lambda), 1(=\lambda\cdot1), 01(=0\cdot1), 011(=0\cdot11), 001(=00\cdot1), 0011(=00\cdot11), ...\}\\ &= \{ 0^i1^j: i,j \geq 0\}\\ \end{align} $$

So your solution is basically right, just not complete, because you need to find a compact way to represent the infinite set in a finite way. The dots are not good enough, because the leave room for interpretation.