2

Let p be the statement: 'If n is an odd number then 4n-1 is a prime' Find the negation of p.

My answer would intuitively be 'If n is an even number then 4n-1 is a prime' but I have the doubt that a possible answer might be 'If n is an odd number then 4n-1 is not a prime'.

I would appreciate it if anybody could help. Thank you very much!

Eli
  • 33

2 Answers2

1

The converse of $p \to q$ is $q \to p$. In your case, $p$ is "$n$ is odd" and $q$ is "$4n-1$ is prime". So the converse is

"If $4n-1$ is prime, then $n$ is odd. "

By the way, the first statement is not true; neither is the second, as $n = 3/4$ shows.

John Hughes
  • 93,729
0

You asked for the negation in the title, but the converse in the body of the question, so it's not clear which you mean. Here are the definitions of each:

For an implication $p \implies q$ (i.e., "If $p$, then $q$"), the converse is the implication $q \implies p$ (i.e., "If $q$, then $p$").

The negation is the "not" statement, and for an implication $p \implies q$, the negation is $p \land (\sim q)$, i.e., "$p$ and not $q$".

Can you figure out the answer now based on your original implication?

layman
  • 20,191
  • Thank you very much, I guess I confused the negation with the converse. This leads to a second question. So, if I understood correctly, if I were asked for the negation of the statement 'if n is an odd number then 4n-1 is a prime'

    the answer would be if n is an odd number then 4n-1 is not a prime?

    Thanks

     – Eli Dec 19 '16 at 14:31
  • @Eli No, you've made a very common but important mistake. Remember the following sentence: The negation of an "if, then" statement is NOT an "if, then" statement. The negation of the statement "if $p$, then $q$" is the statement "$p$ and not $q$". In symbol, we write $\sim(p \implies q)$ is $p \land (\sim q)$. So the negation of the statement "if it rains today, then I am a monkey" is the statement "it rains today and I am not a monkey". Does that make sense? – layman Dec 19 '16 at 14:35
  • 1
    Yes, thank you. Does this mean the answer is 'n is an odd number and 4n-1 is not prime'? – Eli Dec 19 '16 at 14:40
  • @Eli That's right. Good job! :) – layman Dec 19 '16 at 14:42