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Let's break down the problem step by step:

We are given a set \( A = \{ x \in \mathbb{N} \mid x \leq 5 \} \), meaning \( A = \{1, 2, 3, 4, 5\} \), and asked to obtain the solution set, domain, and range for the following relations \( R: A \to A \), with the specific rules of correspondence for each relation:

### Relation \( R_1: x + y < 4 \)

- **Domain**: We need to find all pairs \( (x, y) \in A \times A \) such that \( x + y < 4 \).
- **Solution Set**: 
  - \( (1, 1), (1, 2), (2, 1) \)
- **Range**: \( y = 1, 2 \)

### Relation \( R_2: x - y \) es un número par

- **Domain**: We need to find all pairs \( (x, y) \in A \times A \) such that \( x - y \) is even.
- **Solution Set**:
  - \( (1, 1), (2, 2), (3, 1), (3, 3), (4, 2), (4, 4), (5, 1), (5, 3), (5, 5) \)
- **Range**: \( y = 1, 2, 3, 4, 5 \)

### Relation \( R_3: y = x^2 \)

- **Domain**: We need to find all pairs \( (x, y) \in A \times A \) such that \( y = x^2 \).
- **Solution Set**:
  - \( (1, 1), (2, 4) \)
- **Range**: \( y = 1, 4 \)

### Relation \( R_4: 2x + y = 8 \)

- **Domain**: We need to find all pairs \( (x, y) \in A \times A \) such that \( 2x + y = 8 \).
- **Solution Set**:
  - \( (1, 6), (2, 4), (3, 2) \) → but only pairs where \( y \in A \) count, so:
  - \( (2, 4), (3, 2) \)
- **Range**: \( y = 2, 4 \)

### Relation \( R_5: x + y \) es un número primo

- **Domain**: We need to find all pairs \( (x, y) \in A \times A \) such that \( x + y \) is a prime number.
- **Solution Set**:
  - \( (1, 1), (1, 2), (1, 4), (2, 1), (2, 3), (3, 2), (3, 4), (4, 1), (4, 3), (5, 2) \)
- **Range**: \( y = 1, 2, 3, 4 \)

### Relation \( R_6: x \) es divisible por \( y \)

- **Domain**: We need to find all pairs \( (x, y) \in A \times A \) such that \( x \) is divisible by \( y \).
- **Solution Set**:
  - \( (1, 1), (2, 1), (2, 2), (3, 1), (4, 1), (4, 2), (4, 4), (5, 1) \)
- **Range**: \( y = 1, 2, 4 \)

### Relation \( R_7: x^2 + y^2 \) es un cuadrado perfecto

- **Domain**: We need to find all pairs \( (x, y) \in A \times A \) such that \( x^2 + y^2 \) is a perfect square.
- **Solution Set**:
  - \( (1, 3), (3, 1) \)
- **Range**: \( y = 1, 3 \)

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Do you want to go deeper into any specific relation or need clarification on any of the solutions? 

Here are five related questions to expand on these concepts:
1. How do we find the domain and range for more complex relations?
2. What properties help identify if \( x + y \) is prime more quickly?
3. Can you explain how to identify perfect squares from sums of squares?
4. How can divisibility rules simplify finding solution sets?
5. How would the solutions change if the set \( A \) were expanded?

**Tip**: Always check both domain and range constraints when solving relations between sets to ensure valid solutions.

Given the set A={x ∈ ℕ | x ≤ 5}, find the solution set, domain, and range of the following relations R=A → A with their specified correspondence laws: R1: x+y < 4, R2: x-y is an even number, R3: y = x^2, R4: 2x+y = 8, R5: x+y is a prime number, R6: x is divisible by y, R7: x^2 + y^2 is a perfect square.

Dive into the fascinating world of set relations with constraints on sums, differences, primes, and squares. Discover the solution sets, domains, and ranges for relations within the set A={1,2,3,4,5}, covering concepts like prime numbers, perfect squares, and divisibility.

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