How to show whether a statement is true or false(Example question inside)?

Question

So I'm reading How to Read and Do Proofs by Solow and I'm on the exercises now. So far it has been good but I'm stuck on how to answer a question. There are no answers for even numbered questions in the book. I did the question but I don't know how to clearly explain and represent my answer. Could someone please guide me on how to provide a clean and elegant solution.

Question:

Suppose someone says to you that the following statement is true:"If Mr.Smith wins the election, then you are your own child." Using Table 1 on page 6(It's just a truth table A=>B), did Mr.Smith win the election? Why or why not? Explain.

Here is what I have done so far:

Hypothesis: Mr. Smith wins the election Conclusion: you are your own child.

Let the Hypothesis be denoted by A, and the conclusion by B.

 A     |      B      |  A=>B 

 T     |      T      |   T 

 T     |      F      |   F

 F     |      T      |   T

 F     |      F      |   T

Now, I know that there are 3 possibilities for the statement to be true, but how do I explain it? And the Why or Why not?

A humble request: Please, try to be very explicit in the explanation. I'm self-learning this and I would like a clear answer if possible.

Thanks.

Answer 1

After re-reading the question it stipulates that the implication is true so by a process of elimination you can work it out so this is just a follow up to my comment.

I would say he didn't win the election because obviously B must be false and thus that restricts the truth table down to only those two cases that make B false (lines 2 and 4 in your table). One of those cases is a Truth implies a False (T=>F) which makes the implication false which as stated in the question is true. Thus all that's left is the case F=>F which means A must be false if B is false given that the implication is true.