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The image contains a set of exercises about cardinality, which is a key concept in set theory and deals with comparing the sizes of sets. I'll summarize the questions in English and provide some context for each:

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### Task 4: Cardinality, Comparison of Set Sizes

1. **Prove that the following sets are countable**: 
   - The integers \( \mathbb{Z} \), 
   - The Cartesian product \( \mathbb{N} \times \mathbb{N} \), 
   - The rationals \( \mathbb{Q} \),
   - The set of all finite sequences of natural numbers,
   - The set of all finite subsets of natural numbers,
   - The set of algebraic numbers \( \mathbb{Q}[x] \).

2. **Prove that the following sets are uncountable** (continuum cardinality):
   - The power set \( P(\mathbb{N}) \),
   - The set of all functions from \( \mathbb{N} \) to \( \mathbb{N} \),
   - The real numbers \( \mathbb{R} \),
   - Any open interval in \( \mathbb{R} \),
   - The set of all points in the plane,
   - The set of all sequences of real numbers,
   - The set of all spheres in 3D space,
   - The set of all straight lines in the plane,
   - The set of all continuous functions on \( \mathbb{R} \).

   **(b)** Prove that there exists a transcendental number. Which numbers are more numerous, algebraic or transcendental, and why?

3. **Injections and bijections**:
   - (a) Prove that if \( X \neq \emptyset \), the existence of an injection from \( X \) to \( Y \) is equivalent to the existence of a surjection from \( Y \) to \( X \).
   - (b) Prove that the union of any sequence of countable sets is countable.

4. **Prove that the following sets are finite or countable**:
   - Any set of pairwise non-intersecting intervals on the real line,
   - Any set of pairwise non-intersecting open spheres in 3D space,
   - Any set of pairwise non-intersecting closed letters on a plane,
   - The set of all points of discontinuity of any monotonic function on \( \mathbb{R} \).

5. **Prove the following**: 
   - A set is finite if and only if any injection from the set into itself is a bijection,
   - The cardinality of any finite set is less than the cardinality of any infinite set,
   - The union of any infinite set with a finite set has the same cardinality as the infinite set,
   - The union of any infinite set with a countable set is uncountable,
   - There is no largest infinite cardinality.

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These questions are designed to deepen your understanding of countable and uncountable sets, injections, surjections, and bijections, and their relationships in set theory.

Would you like to explore the solutions to these questions in detail or focus on a specific part? 

### 5 Related Questions:
1. How can we prove that the set of rational numbers is countable?
2. Why is the power set of natural numbers uncountable?
3. What are algebraic and transcendental numbers, and how are they different?
4. How can we use injections and surjections to compare the sizes of sets?
5. What is the cardinality of the real numbers, and how does it compare to the natural numbers?

### Tip: 
When dealing with countability, one powerful technique is to show that a set can be listed in a sequence or mapped to the natural numbers, establishing a bijection or injection.

The image contains a set of exercises about cardinality, which is a key concept in set theory and deals with comparing the sizes of sets. The exercises include proving that certain sets are countable or uncountable, comparing algebraic and transcendental numbers, and discussing injections and surjections.

Explore set theory concepts such as countability and uncountability, cardinality of algebraic and transcendental numbers, and the use of injections and surjections to compare sets. This exercise is ideal for understanding key theorems like Cantor's theorem and the diagonalization argument.

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