1. Let A {p^2/q: p, q ∈ N and q≠ 0}. Prove or disprove that A = is countable.

To determine whether the set \( A = \left\{\frac{p^2}{q} : p, q \in \mathbb{N} \text{ and } q \neq 0 \right\} \) is countable, we need to understand the definition of a countable set.

### Definition of Countable Sets

A set is called **countable** if there exists a **bijection** (one-to-one correspondence) between the set and the set of natural numbers \( \mathbb{N} \). A set is countable if it is either **finite** or **countably infinite** (i.e., it can be put in a one-to-one correspondence with the natural numbers).

### Analyzing the Set \( A \)

The set \( A \) is defined as all fractions of the form \( \frac{p^2}{q} \) where \( p \) and \( q \) are natural numbers (i.e., \( p, q \in \mathbb{N} \) and \( q \neq 0 \)). We want to check whether this set is countable.

#### Steps to Prove \( A \) is Countable

1. **Map \( p^2 \) and \( q \) to \( A \):**

   We can represent any element of \( A \) by choosing values for \( p \) and \( q \). For every \( p \in \mathbb{N} \), \( p^2 \) is a natural number, and for every \( q \in \mathbb{N} \), \( \frac{p^2}{q} \) is a rational number.

2. **Consider the sets of indices \( p \) and \( q \):**

   Both \( p \) and \( q \) range over the natural numbers \( \mathbb{N} = \{1, 2, 3, \ldots\} \).

3. **Construct a function from \( \mathbb{N} \times \mathbb{N} \) to \( A \):**

   Define a function \( f: \mathbb{N} \times \mathbb{N} \to A \) by 
   \[
   f(p, q) = \frac{p^2}{q}.
   \]
   This function maps pairs of natural numbers \((p, q)\) to elements of \( A \).

4. **Show that the set \( \mathbb{N} \times \mathbb{N} \) is countable:**

   The Cartesian product \( \mathbb{N} \times \mathbb{N} \) is countable. This can be shown by arranging the pairs \((p, q)\) in a grid and enumerating them in a diagonal manner:
   \[
   (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1), \ldots
   \]
   This enumeration shows a bijection between \( \mathbb{N} \times \mathbb{N} \) and \( \mathbb{N} \), hence \( \mathbb{N} \times \mathbb{N} \) is countable.

5. **Conclude that \( A \) is countable:**

   Since there is a function \( f \) from \( \mathbb{N} \times \mathbb{N} \) (which is countable) onto \( A \), the image of \( \mathbb{N} \times \mathbb{N} \) under \( f \), which is \( A \), must also be countable.

### Conclusion

The set \( A = \left\{\frac{p^2}{q} : p, q \in \mathbb{N} \text{ and } q \neq 0 \right\} \) is countable because it can be put into a one-to-one correspondence with the set of natural numbers \( \mathbb{N} \).

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Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How can we prove that the set of rational numbers \( \mathbb{Q} \) is countable?
2. What is the difference between a countable and uncountable set?
3. How does the diagonal argument show that the real numbers are uncountable?
4. Can a set that is a subset of a countable set be uncountable?
5. What are some examples of uncountable sets besides the real numbers?

**Tip:** When proving countability, try to construct a function that clearly maps elements from one set to another to establish a bijection.

Explore the proof of countability for the set A = {p^2/q : p, q ∈ N and q ≠ 0}. Understand the concepts of countable sets, Cartesian products, and rational numbers in mathematics. Suitable for advanced mathematics students.

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