countability

Question

whether the given sets countable or uncountable?

1. the set of all finite partitions of N

2. the set of all non-increasing functions from N to N.

3. the set of all non-decreasing functions from N to N.

here, N is natural numbers.

please give answer with proper explanation, as i already have one word answer for all of the problems above.

Comments

Is N set of all natural numbers here?

---

yes

Answer 1

Natural numbers are countable infinite...and any subset of countably infinite is always countable

So All OPTION are countable.

Comments

No, that's not the case.

2 is countable.

1 & 3 are uncountable.

But I don't know the reason why is it so.