To determine $\text{card}(X)$ (the cardinality of the set $X$), where $X = \{2n : n \in \mathbb{N}\}$, let's analyze the elements of $X$.

Step 1: Define the Set $X$

The set $X$ is defined as: $X = \{2n : n \in \mathbb{N}\}$ This means $X$ includes all elements that are twice a natural number. For example:

• If $n = 1$, $2n = 2$
• If $n = 2$, $2n = 4$
• If $n = 3$, $2n = 6$
• And so forth.

Thus, $X$ represents the set of all positive even numbers: $X = \{2, 4, 6, 8, \dots\}$

Step 2: Determine the Cardinality of $X$

To determine $\text{card}(X)$, we need to assess the number of elements in $X$.

1. Mapping to $\mathbb{N}$: Every element $2n$ in $X$ corresponds uniquely to a natural number $n \in \mathbb{N}$.
2. One-to-One Correspondence: The function $f: \mathbb{N} \to X$ defined by $f(n) = 2n$ is a one-to-one correspondence between $\mathbb{N}$ (the set of natural numbers) and $X$ (the set of even natural numbers).
3. Countability: Since there is a one-to-one correspondence between $\mathbb{N}$ and $X$, the cardinality of $X$ is the same as the cardinality of $\mathbb{N}$.

Conclusion

The set $X$ has the same cardinality as $\mathbb{N}$, which is $\aleph_0$ (aleph-null), the cardinality of countably infinite sets. Therefore: $\text{card}(X) = \aleph_0$

Proof Summary

1. $X = \{2n : n \in \mathbb{N}\}$ represents the set of positive even numbers.
2. There exists a bijective (one-to-one and onto) function $f: \mathbb{N} \to X$ defined by $f(n) = 2n$.
3. Since $\mathbb{N}$ is countably infinite and there is a bijection between $\mathbb{N}$ and $X$, $X$ is also countably infinite.
4. Thus, $\text{card}(X) = \aleph_0$.

Would you like more details on this, or do you have any questions?

---

Related Questions

1. What is the cardinality of the set of all odd natural numbers?
2. How does bijection help determine cardinality for infinite sets?
3. Can a subset of a countably infinite set ever be uncountable?
4. What is the cardinality of the power set of $\mathbb{N}$?
5. How do we prove two sets have the same cardinality?

Tip

When determining the cardinality of a set, look for a bijection with a well-known set, such as $\mathbb{N}$, to simplify the process.