I have three questions about elementary set teory and i don't figure out how to solve them:
1)Let $X$ a subset of the cardinal number $2^{\aleph_0}$ (seen as an initial ordinal). Is true or false that $X$ or $2^{\aleph_0} \setminus X$ (the complement set of $X$ in $2^{\aleph_0}$) has the order type of $2^{\aleph_0}$ ?
2) Exist a set $X$ such that $X \subseteq X \times X $ and a set $Y$ such that $Y \times Y \subseteq Y$ ??
3)Let b an ordinal number that $\omega^b = b$ (ordinal exponentation). We can conclude that for every $s,t < b$ we have $s+t<b$ ?
Thanks in advance