Problem 1

Prove the following theorem.

Theorem For any k > 0 and alphabet Σ, the language of all strings over Σ of length k is accepted by a nondeterministic finite automaton.

Note 1: If I asked only about $k = 3$ and $\Sigma = \{a,b\}$, you could prove this by drawing a machine. But now I'm asking you to show that for any $k$ and $\Sigma$ there is an NDFA accepting $\{w \in \Sigma^*\ |\ |w| = k\}$. So there is no single NDFA you could draw to show this.
Note 2: This should be simple. Don't make it tricky, and make sure it's short and sweet.

Problem 2

You're probably (I hope) reasonably convinced from the lecture notes that the language $L = \{a^k b^k\ \big|\ 0 \leq k\}$ cannot be accepted by a finite automaton, despite the lack of a true proof. How about the language $L' = \{a^k b^{k \bmod 2}\ \big|\ 0 \leq k\}$ ? Is there an NDFA that accepts this? Prove your answer!