Implication review! I am currently studying for my Discrete Structures final exam, and there is a question I am not sure how to answer...
Question is:

Consider the following implication. 
"If i do not debug the code or someone tests the program then the
  program has problems."
State the following:  1)Inverse 2)Convere 3)Contrapositive

I got the inverse correct, but not the converse and contrapositive. also would someone mind explaining what they mean? Thanks a lot!
 A: Hint:
The converse of an "if P, then Q" statement is "if Q, then P".
The contrapositive of an "if P, then Q" statement is "if not Q, then not P".
Now you should be able to work it out.
A: Let 
$p$: "I debug the code."
$q$: "Someone tests the program."
$r$: "The program has problems."
Then your original statement is $(\neg p \vee q) \rightarrow r$.


*

*The inverse(i.e. put negation both sides): $\neg(\neg p \vee q) \rightarrow \neg r$ , which is equivalent to, by De Morgan's Law, $(p \wedge \neg q) \rightarrow \neg r $
(This inverse says "If I debug the code and someone does not test the program then the  program has problems.")

*The converse(i.e. reverse the arrow): $r \rightarrow (\neg p \vee q)$

*The contrapositive(i.e. both reverse the arrow and put negation on both sides): $\neg r \rightarrow \neg(\neg p \vee q)$, which is equivalent to $\neg r \rightarrow(p \wedge \neg  q)$. Observe  that this statement is indeed equivalent to saying your original statement.
A: If you have a statement of the form "If $p$, then $q$, the converse is the statement "If $q$, then $p$," and the contrapositive is the statement "If not-$q$, then not-$p$."
The contrapositive is always logically equivalent to the original statement, but the converse is not in general. 
