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  1. Why is this wrong? Also I don't understand why others would be correct. Can anyone please explain? - I get this now.

  2. What does contrapositive mean? Isn't the contrapositive of e.g. "if a then b" if not b then not a? But why is this one wrong? enter image description here

user30200
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    By definition, the converse of "if A then B" is "if B then A". Hence the third option is the one you have to choose. Note that the converse of an implication is not its negation, and in general it is not equivalent to it. – Crostul Sep 16 '20 at 17:09

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For a statement "If P, then Q" one has:

  • The contrapositive "If not Q, then not P"
  • The converse "If Q, then P"

The contrapositive is equivalent to the original statement. It is true precisely when the original statement is true, and is false precisely when the original statement is false.

The converse is a different statement. Knowing whether or not the original statement is true tells you nothing about whether or not the converse is true.

Note also that the negation of "$a$ and $b$ are integers" isn't "$a$ and $b$ are not integers"; the negation is "either $a$ or $b$ is not an integer". That's because the negation of "P and Q" is "not P or not Q"; it isn't "not P and not Q".

To see this, write the statement "$a$ and $b$ are integers" as a true conjunction ("P and Q"), which would be "($a$ is an integer) and ($b$ is an integer)". The English has perhaps hidden what the conjunction really is. So its negation is "($a$ is not an integer) or ($b$ is not an integer)".

MPW
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