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The document contains problems related to set theory, relations, and functions. I'll provide an overview and some guidance on how to approach each question:

### 1. Sets, Relations, Functions
#### a) Prove set properties:
- \( A \cup A = A \)
- \( A \cup B = B \cup A \)
- \( (A \cup B) \cup C = A \cup (B \cup C) \)
- \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
- \( A \cap B = B \cap A \)
- \( (A \cap B) \cap C = A \cap (B \cap C) \)

These are classic set identities, such as idempotence, commutativity, associativity, and distributivity.

#### b) Prove the operation \( A \Delta B = (A \setminus B) \cup (B \setminus A) \):
- Commutativity: \( A \Delta B = B \Delta A \)
- Associativity: \( (A \Delta B) \Delta C = A \Delta (B \Delta C) \)
- Distributivity of \( \Delta \) with respect to intersection:
  
Verify these identities using properties of sets, such as distributivity of union and intersection, and properties of set difference.

### 2. Relations
#### a) Prove that the set \( (x, y) = \{ \{x\}, \{x, y\} \} \) behaves as an ordered pair.
Show that \( (x_1, y_1) = (x_2, y_2) \) implies \( x_1 = x_2 \) and \( y_1 = y_2 \).

#### b) Verify if the composition of binary relations is associative and commutative.
Determine if the composition operation forms a group structure under certain conditions.

### 3. List ordered pairs for sets from \( \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C} \) that allow injections from one to the other.

### 4. Bijection Problems
Determine if there is a bijection between several pairs of sets defined using Cartesian products or power sets.

### 5. Inductive Sets
#### a) Investigate \( x' = x \cup \{x\} \) and find \( P^4(\emptyset) \), where \( P \) denotes the power set.

#### b) Discuss whether an inductive set can be finite.

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Would you like detailed solutions for any specific part of these questions? Let me know if you need more help. Here are five related questions to deepen understanding:

1. How can De Morgan's laws be used to prove set equalities?
2. What is the significance of the composition of relations being associative but not commutative?
3. How does one formally define an injection between infinite sets like \( \mathbb{N} \) and \( \mathbb{Q} \)?
4. What properties must a function satisfy to establish a bijection between two sets?
5. How does transfinite induction differ from standard mathematical induction?

**Tip:** For proving set equalities, Venn diagrams can be a useful visual aid to understand the relationships.

Solve problems related to sets, relations, and functions. Examples include proving set identities, verifying properties of relations, and analyzing bijections and inductions between sets.

Explore advanced problems in set theory, including proofs of set identities, properties of relations, and bijections between sets. Also, examine the concept of inductive sets and the operations on them.

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