About an implication in logic or mathematics.

Question

The following is a question in an entrance examination of a Japanese university.

A quadrilateral $ABCD$ is inscribed in a circle with a radius of $65/8$. The perimeter of this quadrilateral is $44$ and the lengths of $BC$ and $CD$ are both $13$. What are the lengths of the remaining two sides $AB$ and $DA$?

The answer is $AB = 14, DA = 4$ or $AB = 4, DA = 14$.

I think if such a quadrilateral did not exist, this problem would be a very bad problem. So I think we must check the existence of such a quadrilateral. I asked a man about my question. He said $A \implies B$ is true even if $A$ is not true.

Which is correct, me or him?

Answer 1

The way I interpret A$\implies$B is true even if A is not true is, the sum of the two sides is $18$ whether or not the individual sides are $4$ and $14$ which is a correct statement but does not answer the question as to whether they are in fact $4$ and $14$. In this respect, you are correct to want to confirm these values.

Answer 2

The man you talked to is right, and you are wrong.

Logic and mathematics do not deal with what is actually true, but asks: if something is true, then what follows? This is what we mean by the logical implication, for which indeed we use $A \Rightarrow B$. That is, $A \Rightarrow B$ is the case if and on if: $B$ logically follows from $A$. And given the standard axioms of mathematics involving the objects involved in this question, the answer is:

Yes! If we assume that:

A quadrilateral $ABCD$ is inscribed in a circle with a radius of $65/8$. The perimeter of this quadrilateral is 44 and the lengths of $BC$ and $CD$ are both $13$.

then it follows that:

the lengths of the remaining two sides $AB$ and $DA$ is $AB=14,DA=4$ or $AB=4,DA=14$.