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.