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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?

Anonymous
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tchappy ha
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    If $B$ is true, $A\Rightarrow B$ is true. – Javi Jun 04 '18 at 22:40
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    Personally, I'm more inclined to wonder why it even mentions a rectangle in the first place. –  Jun 04 '18 at 22:42
  • Sorry, my english is very bad. Not a rectangle.I mean a quadrilateral. – tchappy ha Jun 04 '18 at 22:44
  • But how's the question related to the quadrilateral? – Javi Jun 04 '18 at 22:48
  • I would guess that there are theorems that will tell us what sort of quadrilateral has a circumcircle. The problem here is given as a premise, say $A$ (the perimeter and various lengths given). Assuming this, i.e. assuming $A$ (not claiming it's true), we have solved the problem and gotten that from $A$ we get $B$ (the solution, using know, valid results). The modus ponens could then be used to claim that $B$ is true, provided we knew $A$ is true. – AnyAD Jun 04 '18 at 23:11
  • The man is correct. Please see this post – amWhy Jun 04 '18 at 23:11
  • By definition $a\to b$ is true, unless $a$ is true and $b$ is false. We can also put that as follows: $a\to b$ is true whenever $a$ is false, or $b$ true, (or both $a$ is false and $b$ is true). – amWhy Jun 04 '18 at 23:14
  • When $a$ is false, the fact that $a\to b$ is true is called a vacuous truth – amWhy Jun 04 '18 at 23:17
  • Thank you very much, Javi, Saucy O'Path, AnyAD, amWhy. – tchappy ha Jun 04 '18 at 23:34
  • A real number $x$ satisfies $x^2 = -1$. What is the value of $x^4$? We cannot calculate $x^4$ because such a real number $x$ does not exist. – tchappy ha Jun 04 '18 at 23:34
  • As you say, for the question to be good, the quadrilateral needs to exist. By the logic of implication, if the quadrilateral doesn't exist then any answer would be correct. As the person answering the question, though, this isn't your problem. – Derek Elkins left SE Jun 05 '18 at 00:19
  • The idea is called "The principle of explosion", also sometimes "vacuous implication". It tends to confuse a lot of those who are new to formal logic, but it is a convention you are probably already using without knowing it. – DanielV Jun 05 '18 at 00:38

2 Answers2

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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 $AB=4,DA=14$.

Bram28
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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.

Phil H
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