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### 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.

Determine whether the two sets have the same cardinality and decide whether each of the given sets is finite, infinite, countable, or uncountable.

Determine the cardinality between sets such as natural numbers, integers, and subsets. Learn about finite, infinite, countable, and uncountable sets in this set theory problem, suitable for college students or advanced high school learners.

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