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Homework Help: Proofs help

  1. Jun 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that for every two distinct real numbers a and b, either (a+b)/2>a or (a+b)/2>b

    2. Relevant equations

    3. The attempt at a solution
    if two distinct numbers a and b then (a+b)/2>a
    Since a≠b and a,bεR, (a+b)/2>a=a+b>2a=b>a. Therefore (a+b)/2>a if b>a.
    if two distinct numbers a and b then (a+b)/2>b
    Since a≠b and a,bεR, (a+b)/2>b=a+b>2b=a>b.
    Therefore (a+b)/2>b if a>b.

    would this suffice as a proof or no?
    Last edited: Jun 14, 2012
  2. jcsd
  3. Jun 14, 2012 #2
    Something is missing. I think you ought to state that a≠b implies either a>b or b>a
    and then show that
    a>b is an equivalent statement to (a+b)/2>b
    b>a => (a+b)/2>a

    Would you have to prove that a≠b => a>b or b>a?
  4. Jun 14, 2012 #3


    Staff: Mentor

    No. You're assuming part of what you need to prove.
    Also, this makes no sense: (a+b)/2>a=a+b>2a=b>a
    You're saying that (a + b)/2 > a, which equals a + b, which is greater than 2a, which equals b, which is greater than a. The problem is that you are apparently connecting inequalities (such as (a+b)/2>a and a + b > 2a), with =. That's not the right symbol. Equations and inequalities aren't equal to anything; they might be equivalent, or one might imply another, but they're not equal.
    Last edited: Jun 14, 2012
  5. Jun 14, 2012 #4
    yeah i knew this was wrong. I think the problem is that im not approaching the conclusion of the result correctly. let me try this again
  6. Jun 15, 2012 #5


    Staff: Mentor

    Break it up into two cases, along the lines of what cryora suggests.

    Case 1: Suppose a < b.
    Show that (a + b)/2 > a.

    Case 2: Suppose that b < a.
    Show that (a + b)/2 > b.
  7. Jun 15, 2012 #6
    proof: Since a and b are distinct numbers this implies that either a>b or b>a

    Case 1: Let a>b. Then a/2>b/2. a/2+b/2>2b/2. (a+b)/2>b. Since a>b then (a+b)/2>b.

    Something like this. Damn proving is tough. Has anyone used a transition to advanced math by chartrand? Im trying to learn how to do proofs from this book. I can do about 70%-80% of the problems but some of them are tricky like this one. Anyone have other books that are good at showing different kinds of proofs that I can use to supplement this book?
  8. Jun 15, 2012 #7


    Staff: Mentor

    Here are a couple that I think would be helpful.
    How to Read and Do Proofs (http://books.google.com/books/about/How_to_read_and_do_proofs.html?id=K3itQwAACAAJ)
    The Nuts and Bolts of Proofs (https://www.amazon.com/Nuts-Bolts-Proofs-Fourth-Edition/dp/0123822173)
  9. Jun 15, 2012 #8
    ok thanks. I'll look into these books.
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