# Inverse, Converse and contraposition of statement?

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

$\neg Rich \Rightarrow \neg Happy$

Converse

$\neg Happy \Rightarrow \neg Rich$

Inverse

$Rich \Rightarrow Happy$

Contrapositive

$Happy \Rightarrow Rich$

(2) Statement

$\neg Stop \Rightarrow \neg Shot$

Converse

$\neg Shot \Rightarrow \neg Stop$

Inverse

$Stop \Rightarrow Shot$

Contrapositive

$Shot \Rightarrow Stop$

Is this true?

UPDATE

(2) Statement

$\neg Stop \Rightarrow Shot$

Converse

$Shot \Rightarrow \neg Stop$

Inverse

$Stop \Rightarrow \neg Shot$

Contrapositive

$\neg Shot \Rightarrow Stop$

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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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