Two questions I encountered in my last Set Theory HW.
1) Let T be a set of all Binary sequences that do not contain 2 consecutive zeros (ex. $100111010\notin T$). Let B be a set that contains all Binary sequences. Do B and T have the same Cardinality? Does T have the same Cardinality as the set that contains all natural series ($|T|=|\mathbb{N}^\mathbb{N}|$)?
To the first question I answered yes since:
Each binary number can be view as a natural number , B is an infinite subset of $\mathbb{N}$ thus $|B|= \aleph_0$. I very similarly take care of T and prove $|T|=\aleph_0$.
To the seconed I answered no since Cantor's Diagonal Argument proves $|\mathbb{N}|<|\mathbb{N}^\mathbb{N}|$.
Is this the way to go about it? am I correct?
2) Prove $|X|<|\{0,1,2\}^X|$. I know Cantor's Theorem gives $|X|<|\{0,1\}^X|$ but I don't know how to proceed.