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Suppose a and b are positive integers. Consider the following statement P:

If 4 ∤ a and 4 ∤ b then 4 ∤ ab. Write down the converse and contrapositive of P. Which of the three statements — P itself, the converse of P, and the contrapositive of P — are true and which false. For the true statements provide a proof, and for the false statements a counterexample.

What I got is the converse of P is 4∤ a.b then 4 ∤ a and 4∤ b for the contrapostive of P is 4 divides a and 4 divides b, then 4 divides a.b

So are the true statments P and the contrapositive of P? or am I not reading the statement properly??

Edit, There was a mistake with the contrapositive statement. Is the new one If 4 Divides a.b then 4 divides a or 4 divides b?

Would this change which statement is true?

HasBeen22
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  • Remember the negation of "p and q" is "not p or not q." – User203940 Nov 01 '23 at 13:52
  • You have errors. Recall your definitions and your logic laws. Your converse is correct of $P$, but your contrapositive is not for multiple reasons. Recall that the contrapositive of $A\implies B$ would be that $\neg B\implies \neg A$. Recall also that $\neg (p\wedge q) = (\neg p)\vee (\neg q)$ – JMoravitz Nov 01 '23 at 13:53
  • As for whether or not $P$ or it's contrapositive is true... note that $4$ does not divide $6$ and that $4$ does not divide $2$ but $4$ does divide $6\times 2$. Note also that the truth of an implication $P$ is always equivalent to the truth of the contrapositive of $P$ – JMoravitz Nov 01 '23 at 13:54

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You've mostly understood the question but you need to remember that the contrapositive of a statement $P \implies Q$ is $\neg Q \implies \neg P$. For example If my statement is "If a shape is a square then it is a rectangle" the contrapositive is "If a shape is not a rectangle then it is not a square".

Neeraj Tatikola
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