elementary set theory - Are the following sets isomorphic or have the same order type?

Are the following sets isomorphic or have the same order type ?

1. $(\mathbb R, \le) ,\ (\mathbb R,\ge)$


2. $(\mathbb Q, \le), \ (\mathbb R, \le) $


3. $(\mathbb N,\ge ), \ (\mathbb N,\le)$


4. $\omega+\omega, \ \omega\cdot\omega$

So in order to show they're isomorphic I need to find an injective function.

1. They are, there is no maximal/minimal element, both have the same cardinality. the function is $f(x)=x $.




2. No, they have different cardinality (is that enough?).




3. I think they aren't since one have a minimal element while the other does not.




4. Both have the same cardinality but the elements are different: one is {1,2,3...1',2'...} the other is {(0,1),(0,2)...} but I think there could be an injection.

Answer

1. You need $f(x)=-x$ instead.




2. Yes. So there cannot be any bijection, much less isomorphism.




3. You're correct.




4. Simpler argument appear to be $\omega+\omega=\omega\cdot2<\omega\cdot\omega$