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Math Help - Negating implications

  1. #1
    Newbie
    Joined
    Mar 2009
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    Negating implications

    Hello all,

    Please help me find the negation of the following:

    1)If a>2 and b<-5, then |a|+ |b| >=7

    So, I think this can be rewritten as A= a>2 and b<-5; B=|a|+ |b| >=7
    and thus, a>2 and b<-5 and |a| + |b|<7

    Though, I'm also confused cause it might be:

    If a<2 and b>5 OR |a| + |b| >=7.....so as you guys can tell i'm throughly confused.


    2) If the numbers a and b are primes, then the number a+b is composite.

    Sort of having the same problem as above.

    Thanks in advance.
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  2. #2
    Junior Member
    Joined
    Oct 2008
    Posts
    38
    Quote Originally Posted by jusstjoe View Post
    Hello all,

    Please help me find the negation of the following:

    1)If a>2 and b<-5, then |a|+ |b| >=7
    Couple of things to lay the ground work.


    • A \implies B \iff \neg A \lor B
    • \neg (A \land B) \iff \neg A \lor \neg B
    • \neg (A \lor B) \iff \neg A \land \neg B
    • \neg (\neg A) \iff A

    Your statement is of the form

    A \land B \implies C

    You want to find,

    \neg (A \land B \implies C)

    Apply the rules above.

    2) If the numbers a and b are primes, then the number a+b is composite.
    Suppose a \neq b \neq 2, then a and b are odd.

    For some  n,m \in \mathbb{N} then,

    a = 2n-1

    b = 2m-1

    This implies,
    a+b = 2(n+m) - 2 = 2 (n+m-1)

    Therefore a+b is even and not prime.

    Now, if a or b is 2 then...this is not true. Consider the counter example with
    a=2
    b=3
    then a+b=5 is prime (not composite)
    Last edited by n0083; March 23rd 2009 at 10:47 PM.
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