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I need help with the negation in discrete math

The question is : Negate the statement and express your answer in a smooth english sentence. Hint first rewrite the statement so that it does not contain an implication. The statement is: If the bus is not coming, then I cannot get to school.

My solution is: 1) The bus is coming, I can get to school. therefore 2) I can get to school because the bus is coming.

I got zero on it what did I went wrong? or I totally screwed up and misunderstand? Thanks :)

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2 Answers

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The negation of :

$\lnot p \to \lnot q$

is :

$\lnot p \land q$.

Thus, the answer is :

"The bus is not coming and I can get to school".

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Here is what you should do in logic:

$$\eqalign{ & \neg \left( {P \to R} \right) \leftrightarrow \neg \left( {\neg P \vee R} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Conditional\,\,Disjunction \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ \leftrightarrow \neg \neg P \wedge \neg R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Demorgan's\,Law \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ \leftrightarrow \,\,\,\,\,\,\,\,\,P \wedge \neg R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Double\,Negation \cr} $$

your $P$ and $Q$ are

$P=$the bus is not coming
$Q=$I cannot get to school

so the negation is

the bus is not coming and I can get to school

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