Cardinal numbers

Question

Suppose $m, n$ are infinite ordinal numbers. $$a) m=n → |m|=|n|$$ $$b)|m|=|n| →m=n$$ $$c)m<n→ |m|<|n|$$ $$d)|\max{(m,n)}|< |m|+|n|$$ $$e)|m|<|n| →|m|^{|n|}<|n|^m$$

Which of the above statements are true? (a) looks true but I do not know the way to work it out. Please help

Answer 1

Hints:

1. Recall that $|x|=|y|$ is an equivalence relation.
2. Recall that $|\Bbb N|=|\Bbb N\setminus\{0\}|$, so adding or removing one element does not change the cardinality. You said that you know what is a successor ordinal, this should give you a counterexample.
3. The counterexample to the previous statement should give you a counterexample to this one as well.
4. Recall that addition of cardinals satisfy $|a|+|b|=\max\{|a|,|b|\}$.
5. Consider $\omega$ and $\frak c$ (as suggested by Chris Eagle in the comments).