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If you have proven that the statement is True, then you can say that the contrapositive of the statement is also True. 
Now what about the converse and inverse of that same statement is is True or False? 
Or do you have to prove the newly created statements to determine their outcome.

Are converse and inverse false is the statement is true?
Can you say anything about the converse and inverse of a statement?

4 Answers4

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"If I am Bill Clinton, then I am a man." Presumably you agree that this statement is true.

Now, what about the converse? "If I am a man, then I am Bill Clinton."

Daniel McLaury
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If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

But there is NO relation between the statement and it's coverse and inverse.

Counterexamples

  • Statement :If two angles are congruent, then they have the same measure.(TRUE)
  • Converse :If two angles have the same measure, then they are congruent.(TRUE)
  • Inverse :If two angles are not congruent, then they do not have the same measure.(TRUE)
  • Contrapositive:If two angles do not have the same measure, then they are not congruent.(TRUE)
    Another example.
  • Statment : If a quadrilateral is a rectangle, then it has two pairs of parallel sides (TRUE)
  • Converse : If a quadrilateral has two pairs of parallel sides, then it is a rectangle. (FALSE!)
  • Inverse : If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides. (FALSE!)
  • Contrapositive : If a quadrilateral does not have two pairs of parallel sides, then it is not a rectangle.(TRUE)
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In general, no, we cannot say anything about the converse and inverse.

Since you refer to the contrapositive, we can take the statement to be $A\implies B$.

So its contrapositive is $\neg B\implies \neg A$,
its converse is $\neg A\implies \neg B$,
and its inverse is $B\implies A$.

The converse and inverse do not follow from the statement itself; the contrapositive does.

Note that the converse and inverse are contrapositives of each other, so if one is true the other one is. Note too that the converse is the inverse of the contrapositive and the inverse is the converse of the contrapositive.

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Statement: if A then B. Converse: if B then A Inverse: if not A then not B Contrapositive: If not B then not A

It is (reasonably) easy to show that if the "statement" is true then the "contrapositive" is true and vice-versa.

Similarly, the "converse" is true if and only if the "inverse" is true.

But the truth of the "statement" tells us nothing about whether the "inverse" and "converse" are true.

user247327
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