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I'm a little bit confused by the contraposition. Suppose we have a statement: "If number can be divided by 3 ($P$), Then it can also be divided by 9 ($Q$)"

In a book "Discrete mathematics with applications" there is an exercise where the reader must provide a contrapositive to this statement. In the answers section the correct answer is this: "If number can be divided by 9, then it can be divided by 3". But isn't this a simple conversion $Q\rightarrow P$?

By the law of contrapositive: $P\rightarrow Q\equiv\neg Q\rightarrow\neg P$ which can be translated as "If number can not be divided by 9, then it cannot be divided by 3", which is obviously not true.

Maxim Nikitin
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Yolanda
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    Yes, you are right. If that's what the book says, then the book is wrong. – José Carlos Santos Aug 24 '21 at 07:20
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    The statement "If a number is divisible by $3$, then it is divisible by $9$" is equivalent to its contrapositive "If a number is not divisible by $9$, then it is not divisible by $3$". Of course, both statements are not true. – Matthias Klupsch Aug 24 '21 at 07:24
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    Logical equivalence has nothing to do with the truth of the statement. It just says that the truth of one is equivalent to the truth of the other. – John Douma Aug 24 '21 at 07:36
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    It seems that there is confusion between the terms contrapositive and converse. – user2661923 Aug 24 '21 at 08:01

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"If number can be divided by 3 ($P$), Then it can also be divided by 9 ($Q$)"....

This statement is false, e.g. 6 is divisible by 3, and 6 is not divisible by 9.

By the law of contrapositive: P→Q ≡ ¬Q→¬P which can be translated as "If number can not be divided by 9, then it cannot be divided by 3", which is obviously not true.

Nothing wrong with that. You can take the contrapositive of a false statement. The contrapositive will also then be false.

Dan Christensen
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