The image provided contains the following problem from an exercise sheet:

Problem 13: "Show that the set $E = \{2k : k \in \mathbb{N}\}$ is countably infinite."

Solution Outline:

To show that the set $E = \{2k : k \in \mathbb{N}\}$ is countably infinite, we need to demonstrate two key points:

1. The set $E$ is infinite: This requires us to show that there is no bound on the size of the set, i.e., it has infinitely many elements.

2. The set $E$ is countable: This means that there exists a one-to-one correspondence (a bijection) between the set $E$ and the set of natural numbers $\mathbb{N}$.

Step 1: Showing that $E$ is infinite

The set $E$ consists of even natural numbers, because each element of $E$ is of the form $2k$ where $k \in \mathbb{N}$. As $k$ increases without bound (since $\mathbb{N}$ is infinite), the values $2k$ also increase without bound. Therefore, the set $E$ contains an infinite number of elements, since there is no upper limit on the values of $k$ in $\mathbb{N}$.

Step 2: Showing that $E$ is countable

To show that $E$ is countable, we need to construct a bijection between $\mathbb{N}$ (the set of natural numbers) and $E$ (the set of even natural numbers). Define a function $f : \mathbb{N} \to E$ by the rule:

$f(k) = 2k \quad \text{for all} \quad k \in \mathbb{N}.$

• This function is injective (one-to-one) because if $f(k_1) = f(k_2)$, then $2k_1 = 2k_2$, which implies $k_1 = k_2$.

• The function is also surjective (onto) because for every element $e \in E$, we can find a corresponding $k \in \mathbb{N}$ such that $e = 2k$. Specifically, for any $e \in E$, $e$ must be of the form $2k$ for some $k \in \mathbb{N}$, so $f(k) = e$.

Since the function $f$ is both injective and surjective, it is a bijection between $\mathbb{N}$ and $E$.

Conclusion:

Since there exists a bijection between $\mathbb{N}$ and $E$, the set $E = \{2k : k \in \mathbb{N}\}$ is countably infinite.

Would you like further clarification on any part of this explanation?

Here are five related questions that can extend your understanding:

1. How would you prove that the set of odd numbers is countably infinite?
2. What is the definition of a countably infinite set in mathematical terms?
3. Can you describe another example of a set that is countably infinite?
4. What is the difference between countably infinite and uncountably infinite sets?
5. How can we extend this concept to show that all subsets of natural numbers are countable?

Tip: When proving a set is countable, finding a bijection with $\mathbb{N}$ is a common strategy to establish the countability.