Write the contrapositive of the conditional statement below.
“ n is a prime number implies that n=2 or n is an odd number.”
Your answer must contain a conjunction in its premise.

- anonymous

- schrodinger

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- KingGeorge

So in general, the contrapositive of the statement (\(\,p\implies q\)) is \((\neg \,q\implies \neg \,p\)). In this case, both your p and q are other sentences. Can you tell me what the "p" is in your conditional statement?

- anonymous

n is a prime number

- KingGeorge

Right. And the "q"?

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## More answers

- anonymous

n is an odd number

- KingGeorge

Almost. The "q" in your statement is "n=2 or n is an odd number." Notice that there's an "or" in this statement, so even this can be broken down into something like "q OR r," where q is "n=2", and r is "n is an odd number."

- anonymous

oh i

- anonymous

SEE

- KingGeorge

So the statement "n is a prime number implies that n=2 or n is an odd number" can be rewritten as \[p\implies (q \vee r)\]

- anonymous

if n is not a odd number then n is not a prime number or n = 2

- KingGeorge

You're very close, and certainly have the right idea. The contrapositive of \(p\implies (q\vee r)\) is \(\neg(q\vee r)\implies \neg p\). Can you tell me what \(\neg(q\vee r)\) can be rewritten as?

- anonymous

not sure

- KingGeorge

This is a thing called "De Morgan's Law." It says that \[\neg(q\vee r) \Longleftrightarrow (\neg \,q) \wedge(\neg\,r) \]Can you translate this back into our statements?

- anonymous

im confused

- KingGeorge

So if q is "n=2," what is ~q?

- anonymous

not =2

- anonymous

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- KingGeorge

Right. And if r is "n is an odd number," what is ~r?

- anonymous

n is not an odd number

- KingGeorge

Bingo. So if we then combine those two statements, ~q AND ~r can be written as "n is not 2, and n is not an odd number."
Make sense?

- anonymous

yes

- KingGeorge

Great. Then the contrapositive of your original statement should be
"If n is not 2, and n is not an odd number, then n is not prime."

- anonymous

that the answer it was that simple?

- KingGeorge

That was it. Did it all make sense?

- KingGeorge

You almost had it the first time, you just got a little mixed up with the second half of the statement.

- anonymous

yes but what does it mean by Your answer must contain a conjunction in its premise.

- KingGeorge

That just means that it has to have "and" somewhere in the statement. So the way it's written, it's fine.
If we had written it as "If n is an even integer greater than two, then n is not prime" instead, which says the same thing, it would not be the correct solution because it does not contain "and."

- anonymous

oh i see thanks so much!!

- KingGeorge

You're welcome.

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