Valid or Invalid

"Amy is a student" is true; Ben is a teacher" is true; "Amy came to the school today" is false; Ben came to the school today" is true. Determine whether the sentence in each of the questions is true or false. If it cannot be determined by the above information, just answer "undetermined. So, your answer to each question will be one of "true", "false", or "undetermined. You do not need to explain why. 1- If Ben came to the school today, then either Amy is a student or Ben is not a teacher. 2- Although Ben is not a teacher, he came to the school today; but Amy did not come to the school today although she is a student. 3- Amy is not a student but she came to the school today; and Ben is a teacher but he did not come to the school today. 4- The necessary condition for Amy to come to the school today is that she is a student. 5- Either Ben is not a teacher, or Amy is a student only if she came to the school today. 6- Ben cam to the school today because he is a teacher, but Amy did not come to the school today because she is not a student. 7- Amy is a student and she came to the school today, if and only if Ben is a teacher and he came to the school today. 8- Amy is a student, but she did not come to the school today unless Ben came to the school today