Let's work through the questions visible in the image:

Problem 5: Cardinality of Sets

This question asks you to determine if two sets have the same cardinality (i.e., the same "size" or number of elements, which may include infinite sets). We’ll analyze each pair:

1. The set of even natural numbers and $\mathbb{N}$ (natural numbers):

   • The set of even natural numbers $\{2, 4, 6, 8, \dots\}$ is infinite, and so is $\mathbb{N}$. These two sets have the same cardinality because there is a one-to-one correspondence between them. You can map $n \in \mathbb{N}$ to $2n \in \text{Even Numbers}$.

2. The set of integers $\mathbb{Z}$ and the set of negative integers:

   • The set of integers $\mathbb{Z}$ contains both positive and negative integers, while the set of negative integers is only half of $\mathbb{Z}$. The cardinality of $\mathbb{Z}$ is larger, so these two do not have the same cardinality.

3. The set $\{1, 2, 3, \dots, 210\}$ and $\mathbb{N}$:

   • The set $\{1, 2, 3, \dots, 210\}$ is a finite set of 210 elements, while $\mathbb{N}$ is infinite. Therefore, these two sets do not have the same cardinality.

4. $\mathbb{Z}$ (the set of integers) and $\mathbb{N}$:

   • Although $\mathbb{Z}$ contains both negative and positive integers, the two sets are countably infinite. There is a one-to-one correspondence between them, so they have the same cardinality.

Problem 6: Finite, Infinite, Countable, or Uncountable Sets

This asks whether each set is finite or infinite, and countable or uncountable.

1. $\mathbb{N}$ (the set of natural numbers):

   • This is an infinite set, but it is countable since you can list the natural numbers.

2. The set of odd natural numbers:

   • This is also infinite and countable. There is a one-to-one correspondence between natural numbers and odd natural numbers $n \to 2n - 1$.

3. The set $\{x \in \mathbb{N} : x \mod 2 = 0\}$ (even natural numbers):

   • This is the set of even natural numbers, which is infinite and countable, just like the odd natural numbers.

4. The set of negative integers:

   • This set is also infinite and countable since there’s a one-to-one correspondence with the natural numbers (e.g., $n \to -n$).

Would you like further details or explanations on any of these? Here are some related questions you might explore:

1. What is the concept of bijections in proving same cardinality?
2. Why are countably infinite sets important in mathematics?
3. How do we establish a one-to-one correspondence between $\mathbb{Z}$ and $\mathbb{N}$?
4. What is the difference between countable and uncountable infinity?
5. Can a finite set ever have the same cardinality as an infinite set?

Tip: When comparing infinite sets, focus on finding a mapping that shows whether or not each element of one set can be paired with an element from another set.