Prove or disprove those languages are regular

Question

in my practice for a test I came across this question: prove or disprove that those languages are regular:

I succeeded proving that the second language is nonregular with homomorphism but i'm having trouble with the other 2 -

1. my intuition tellin me that first 1 is Σ∗ but i'm really stuck on how to prove it

2. (not sure why it writes 2 when i'm writin 3 haha) in this one i'm really stuck - my intuition is tellin me again that it will be finite automaton (automata? sorry if my english is not so good) and therefore regular language but how can I prove it? how can I use the fact that |w| is bounded? - i guess I need to build an automaton , maybe with some "flags" but I suck at it (for now) :[.

does ww^R is regular language when |w| is bounded? I've looked a lot (through google in this site) for an answer for this question but I havent find something - only that this language is not regular when |w| is not bounded.

Can I get some hints on both of those languages please?

Answer 1

For number 1, you're entirely correct. Every string in $\Sigma *$ is in this language. Take an arbitrary string of zeroes, ones, and twos, then let $y$ be that string, and $w$ be empty.

For number 2, the standard way I learned to prove it's not regular is with "fooling sets". The Pumping Lemma also works here.

For number 3, consider that there are only finitely many possibilities for $w$. This means you could, in theory, write a regular expression that just listed all of them. So this one is actually regular, through "brute force".