State the truth value of each statement below. Briefly explain why the statement is true
or false.
(a) ∀x ∈ Z, ∃y ∈ Z such that x = y
2
.


Let's break down the statement step by step to determine its truth value.

### Statement:
**∀x ∈ Z, ∃y ∈ Z such that x = y².**

- **∀x ∈ Z**: This means "for every integer \( x \in Z \)."
- **∃y ∈ Z such that \( x = y^2 \)**: This means "there exists an integer \( y \in Z \) such that \( x = y^2 \)," or equivalently, \( x \) is a perfect square.

In simpler terms, the statement says that for every integer \( x \), there exists an integer \( y \) such that \( x \) is the square of \( y \).

### Explanation:

This statement is **false**. Here's why:

- Not every integer \( x \) can be written as the square of some integer \( y \). For example:
  - \( x = 2 \) is not a perfect square because there is no integer \( y \) such that \( y^2 = 2 \).
  - \( x = -1 \) is also not a perfect square since the square of any integer is non-negative, and no integer squared gives a negative number.

Thus, the statement is false because there are many integers \( x \) that cannot be expressed as the square of any integer.

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### Do you want further details or have any other questions?

Here are 5 related questions for further exploration:

1. Can every positive integer be written as a sum of two squares?
2. What is the difference between a perfect square and a perfect cube in number theory?
3. How many perfect squares are there between two consecutive perfect squares?
4. Is there a way to determine if a number is a perfect square without calculating the square root?
5. Can every integer be written as a product of prime factors?

**Tip:** A perfect square is an integer that can be expressed as the square of another integer. Recognizing these can be useful in factorization and simplification problems.

State the truth value of each statement below. Briefly explain why the statement is true or false. (a) ∀x ∈ Z, ∃y ∈ Z such that x = y².

Explore the truth value of a universal statement in number theory, determining whether every integer x can be written as the square of some integer y. This problem helps understand the concept of perfect squares and their properties. Ideal for high school students studying algebra and number theory.

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