1. **Problem statement:** Determine the truth value of the statement $\forall n \exists m (n^2 < m)$ where the domain for all variables is all integers. 2. **Understanding the statement:** - $\forall n$ means "for every integer $n$". - $\exists m$ means "there exists an integer $m$". - The inequality $n^2 < m$ means $m$ is greater than the square of $n$. 3. **Interpreting the statement:** For every integer $n$, there must be some integer $m$ such that $m$ is greater than $n^2$. 4. **Is the statement true?** - Since the integers are infinite and unbounded above, for any integer $n$, we can always find an integer $m$ greater than $n^2$. - For example, choose $m = n^2 + 1$. - This $m$ satisfies $n^2 < m$. 5. **Conclusion:** The statement $\forall n \exists m (n^2 < m)$ is **true**. 6. **Example:** For $n = 3$, $n^2 = 9$, choose $m = 10$ which satisfies $9 < 10$. Final answer: The statement is true, and an example for $n=3$ is $m=10$.