1. **Stating the problem:** We are given custom relational operators with the following meanings: - $AAB$: $A$ is not smaller than $B$ (i.e., $A \geq B$) - $A*B$: $A$ is neither smaller than nor equal to $B$ (i.e., $A > B$) - $A\otimes B$: $A$ is neither bigger than nor equal to $B$ (i.e., $A < B$) - $A+B$: $A$ is not bigger than $B$ (i.e., $A \leq B$) - $A\%B$: $A$ is neither bigger than nor smaller than $B$ (i.e., $A = B$) The statement is: [email protected]%T$ We need to analyze the conclusions: I. $P@T$ II. $Q+T$ III. $P*R$ 2. **Interpreting the statement:** The statement uses operators $@$ and $.$ which are not defined explicitly. Assuming $@$ and $.$ are placeholders or concatenations, the key part is $AR%T$ which means $A$ and $R$ relate by $*$ and $R$ and $T$ relate by $%$. Since $AR%T$ means $R%T$, i.e., $R = T$. 3. **Analyzing conclusions:** - I. $P@T$: Since $@$ is undefined, we cannot confirm this directly. - II. $Q+T$: $Q+T$ means $Q \leq T$. - III. $P*R$: $P*R$ means $P > R$. 4. **Using $R = T$ from $R%T$:** - From $R = T$, $P*R$ is equivalent to $P > T$. 5. **Summary:** - $P*R$ means $P > R$ and since $R = T$, $P > T$. - $Q+T$ means $Q \leq T$. 6. **Final answers:** - Conclusion I ($P@T$) cannot be determined due to undefined operator $@$. - Conclusion II ($Q+T$) means $Q \leq T$. - Conclusion III ($P*R$) means $P > R$ and since $R = T$, $P > T$. Hence, the valid conclusions based on the given information are II and III.