The problem asks us to evaluate the truth value of five quantified statements where the universe of discourse for all variables is the set of all integers ($\mathbb{Z}$). We need to determine which statements are true.

Evaluating Quantified Statements Over Integers

Statement A: $\forall n \exists m (n^2 < m)$

• Meaning: For every integer $n$, there exists an integer $m$ such that $n^2$ is strictly less than $m$.
• Analysis: Choose any integer $n$. Let $m = n^2 + 1$. Since $n$ is an integer, $n^2$ is an integer, and $n^2 + 1$ is also an integer. By definition, $n^2 < n^2 + 1$.
• Conclusion: Statement A is TRUE.

Statement B: $\exists n \exists m (n + m = 4 \land n - m = 2)$

• Meaning: There exist integers $n$ and $m$ that simultaneously satisfy both $n + m = 4$ and $n - m = 2$.
• Analysis: We can solve this system of linear equations.
  • Adding the two equations: $(n + m) + (n - m) = 4 + 2 \implies 2n = 6 \implies n = 3$.
  • Substituting $n = 3$ into $n + m = 4$: $3 + m = 4 \implies m = 1$.
  Since $n = 3$ and $m = 1$ are integers, such integers exist.
• Conclusion: Statement B is TRUE.

Statement C: $\exists n \exists m (n^2 + m^2 = 5)$

• Meaning: There exist integers $n$ and $m$ such that the sum of their squares is 5.
• Analysis: We look for integer pairs whose squares sum to 5. Consider the squares of small integers: $0^2=0$, $1^2=1$, $2^2=4$, $3^2=9$, ... The pair of squares $(1, 4)$ sums to 5 ($1 + 4 = 5$). This corresponds to $n^2=1$ and $m^2=4$ (or vice versa). Integer solutions exist, for example, $n=1$ and $m=2$, since $1^2 + 2^2 = 1 + 4 = 5$. Other pairs like $n=2, m=1$ or involving negative integers also work.
• Conclusion: Statement C is TRUE.

Statement D: $\exists m \forall n (m + n = 0)$

• Meaning: There exists a specific integer $m$ such that for all integers $n$, the equation $m + n = 0$ holds true.
• Analysis: The equation $m + n = 0$ implies $n = -m$. The statement requires that there is one integer $m$ for which *every* integer $n$ equals $-m$. This is impossible, as $n$ can take infinitely many different values, while $-m$ is a single, fixed value for a given $m$. For instance, if $m=1$, we would need $n=-1$ for all $n$, which is false.
• Conclusion: Statement D is FALSE.

Statement E: $\forall m \exists n (m + n = 0)$

• Meaning: For every integer $m$, there exists an integer $n$ such that $m + n = 0$.
• Analysis: Choose any integer $m$. Let $n = -m$. Since $m$ is an integer, its additive inverse, $-m$, is also an integer. The equation $m + (-m) = 0$ is always true.
• Conclusion: Statement E is TRUE.

Final Determination

• Statements A, B, C, and E are true.
• Statement D is false.

Therefore, the set of true statements is {A, B, C, E}.