anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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KingGeorge
  • 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
  • anonymous
n is a prime number
KingGeorge
  • KingGeorge
Right. And the "q"?

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anonymous
  • anonymous
n is an odd number
KingGeorge
  • 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
  • anonymous
oh i
anonymous
  • anonymous
SEE
KingGeorge
  • 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
  • anonymous
if n is not a odd number then n is not a prime number or n = 2
KingGeorge
  • 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
  • anonymous
not sure
KingGeorge
  • 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
  • anonymous
im confused
KingGeorge
  • KingGeorge
So if q is "n=2," what is ~q?
anonymous
  • anonymous
not =2
anonymous
  • anonymous
|dw:1355893931180:dw|
KingGeorge
  • KingGeorge
Right. And if r is "n is an odd number," what is ~r?
anonymous
  • anonymous
n is not an odd number
KingGeorge
  • 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
  • anonymous
yes
KingGeorge
  • 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
  • anonymous
that the answer it was that simple?
KingGeorge
  • KingGeorge
That was it. Did it all make sense?
KingGeorge
  • KingGeorge
You almost had it the first time, you just got a little mixed up with the second half of the statement.
anonymous
  • anonymous
yes but what does it mean by Your answer must contain a conjunction in its premise.
KingGeorge
  • 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
  • anonymous
oh i see thanks so much!!
KingGeorge
  • KingGeorge
You're welcome.

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