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I am trying to break the statements:

"Being rich is necessary for Alex to be happy"(1) and "Stop, or I will shoot!"(2)

(1) Statement

$\neg Rich \Rightarrow \neg Happy$


$\neg Happy \Rightarrow \neg Rich $


$ Rich \Rightarrow Happy$


$Happy \Rightarrow Rich $

(2) Statement

$\neg Stop \Rightarrow \neg Shot$


$\neg Shot \Rightarrow \neg Stop $


$ Stop \Rightarrow Shot$


$Shot \Rightarrow Stop $

Is this true?

I appreciate your answer!


(2) Statement

$\neg Stop \Rightarrow Shot$


$ Shot \Rightarrow \neg Stop $


$ Stop \Rightarrow \neg Shot$


$\neg Shot \Rightarrow Stop $

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up vote 4 down vote accepted

Your answers to $(1)$ are just fine. You seem to understand the relationships between an implication, its converse, inverse, and its contrapositive.

Implication: $A \equiv (p \rightarrow q)$

  • converse of $A$: $\;q\rightarrow p$
  • inverse of $A$: $\;\lnot p \rightarrow \lnot q$
  • contrapositive: $\;\lnot q \rightarrow \lnot p$

Note that $\,A \equiv \text{contrapositive(A)}\;$ and $\;\text{inverse(A)}\equiv \text{converse(A)}$.

$(2)$ Here, your initial translation is incorrect, and as a consequence, so are the converse, inverse, and contrapositives.

Let's look at $(2)$ again.

Stop, or I'll shoot $\iff$ If you don't stop, then I'll shoot.

This can be translated into two equivalent logical statements:

$\text{Stop} \lor \text{Shot}\,\equiv \lnot \,\text{Stop}\rightarrow \text{Shot}\tag{2}$

Now, use what you know about the converse of an implication, the inverse, and the contrapositive to write the corresponding statements to the implication given on the right-hand side of $(2)$

UPDATE: Now you're spot on!

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Thx for your answer! I updated my answer! – Le Chifre Jul 30 '13 at 11:16
And you're spot on! ;-) – amWhy Jul 30 '13 at 11:17
@amWhy: What does it mean by saying Spot on?+ – Babak S. Jul 30 '13 at 19:02
Hi, @Babak: it means: "indeed, you're understanding of the matter, and your answer, are entirely correct." – amWhy Jul 30 '13 at 21:38
@amWhy: It even has a history! +1 – Amzoti Jul 31 '13 at 0:25

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