Let $m$ and $n$ be infinite ordinal numbers
Which of the following is true
a) $m<n \Rightarrow |m|^{|m|}<n^{|n|}$
b) $m+n$= Max{$m,n$}
c) $m=n \Rightarrow |m|=|n|$
d)$|m|=|n| \Rightarrow m=n $
e) Max{$m,n$} <$|m|+|n|$
I could only understand that d) is false due to the cardinality of the two different ordinals can be the same and c is true! Please help on the rest of the statements
(Assuming ordinal operations) For a) note that $\omega < \omega + 1$, and as $\omega^n \le (\omega + 1)^n < \omega^\omega$ holds for all $n$, we have $$\omega^\omega = \sup_n \omega^n \le \sup_n (\omega + 1)^n = (\omega + 1)^\omega \le \omega^\omega$$ so $(\omega + 1)^\omega = \omega^\omega$, showing that a) is wrong.
For b) note that $\omega + \omega = \omega2 > \omega = \max\{\omega, \omega\}$.
For e) finally the strict inequality needn't hold in general, as one can see for example from $$ \omega_1 = \max\{\omega, \omega_1 \} = \omega + \omega_1 $$