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.

Question

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.

kinggeorge · Answer

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 · Answer

n is a prime number

kinggeorge · Answer

Right. And the "q"?

anonymous · Answer

n is an odd number

kinggeorge · Answer

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 · Answer

oh i

anonymous · Answer

SEE

kinggeorge · Answer

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 · Answer

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

kinggeorge · Answer

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 · Answer

not sure

kinggeorge · Answer

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 · Answer

im confused

kinggeorge · Answer

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

anonymous · Answer

not =2

anonymous · Answer

|dw:1355893931180:dw|

kinggeorge · Answer

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

anonymous · Answer

n is not an odd number

kinggeorge · Answer

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 · Answer

yes

kinggeorge · Answer

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 · Answer

that the answer it was that simple?

kinggeorge · Answer

That was it. Did it all make sense?

kinggeorge · Answer

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

anonymous · Answer

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

kinggeorge · Answer

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 · Answer

oh i see thanks so much!!

kinggeorge · Answer

You're welcome.