Assignment 1: Logic Application

I am playing “Guess Your Cards” with Andy, Belle, and Carol. Andy has drawn a 1, 5 and 7, Belle a 4, 5 and 7, and Carol a 2, 4 and 6. No one can see their own cards. Question cards are drawn and asked to help each player deduce what their own cards are.
I believe deductive reasoning would be the logic used to solve this problem; you have the facts in front of you. The deck has only cards with the numbers 1 through 9 on them, you can see the cards already drawn and deduct, by the answers to the questions, exactly what cards you have.
Let’s start with the first question. “Do you see two or more players whose cards sum to the same value?” Andy answers “yes”. Of the cards I can see, no two people have the same sum. I deduce that I must have the second set. Adding the cards of each person, I can see that Andy’s cards equal 13, Belle’s equal 16, and Carol’s equal 12. My set of cards must equal one of these.
Second, “Of the five odd numbers, how many different odd numbers do you see?” Belle answers that she sees all of them. I can only see 1, 5, and 7. Therefore, I must have a 3 and a 9. These are the only two odd numbers that I cannot see.
The statement that Andy knows what cards he has is totally irrelevant to the problem. It has absolutely no bearing on my logic or what cards I have.
Knowing I have a 3 and a 9, I only have to figure out what my last card is. My 3 and 9 add up to 12, therefore, my sum cannot to be equal to Andy or Carol because Andy has 13 and I am below that with only two cards, Carol has 12 and I am equal to that with only two cards. I have to equal 16 so my last card must be a 4. “3+9+4=16”
I came to my conclusion with pure, simple logic and nothing else. The only other way to figure out my cards might be is to guess, and that is very rarely an accurate way to come to a