1. **Problem Statement:** Determine the truth value of the statement $\forall x \exists y (x^2 = y)$ where the domain of $x$ and $y$ is all real numbers. 2. **Understanding the statement:** - $\forall x$ means "for all real numbers $x$". - $\exists y$ means "there exists a real number $y$". - The statement $x^2 = y$ means $y$ is the square of $x$. 3. **Interpreting the statement:** For every real number $x$, there must be some real number $y$ such that $y = x^2$. 4. **Check if the statement is true:** - For any real number $x$, $x^2$ is always a real number (since squaring a real number results in a non-negative real number). - Therefore, for each $x$, we can choose $y = x^2$. 5. **Conclusion:** The statement $\forall x \exists y (x^2 = y)$ is **true** because for every real $x$, there exists a real $y$ equal to $x^2$. **Final answer:** The statement is true.