Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Need help with two simple proofs

  1. Sep 19, 2004 #1
    Here's my problem:
    Provide either a proof or a counterexample for each of these statements.
    a) For all real numbers x and y, if x is greater than 1 and y is greater than zero, then y^x is greater than x.

    My proof:
    Suppose x is some real number greater than 1 and y is some real number greater than 0.
    Suppose x=2 and y=1/4.
    Then y^x=(1/4)^2=1/16 and 1/16=y^x is less than x=2.
    Now suppose that x=3 and y=2.
    Then y^x=2^3=8 and 8=y^x is greater than x=3.
    Hence if x is greater than 1 and 0 less than y less than or equal to 1, then y^x is less than or equal to x.
    But if x is greater than 1 and y is greater than 1, then y^x is greater than x.
    Therefore the statement "if x is greater than 1 and y is greater than 0, then y^x is greater than x" is not true for all real numbers x and y.

    Is this a good proof? How can I improve it or make it clearer?

    b) For integers a, b, c, if a divides bc, then either a divides b or a divides c.

    I'm not really sure where to go with this one, so hints would be welcome.
    I do know that if a divides bc, then bc=ak, where k is a natural number.
    Similarly, a divides b means that b=aj and a divides c means that c=ai, where j and i are also natural numbers.
    Which proof techinique do I use here? contradiction, contraposition, or direct proof?

    Thanks ahead of time,
    eku_girl83
     
  2. jcsd
  3. Sep 19, 2004 #2
    your first proof looks solid ..
    what u have done is simply give a counter example
    u could have chosen some simpler values say,
    x=2 and y=0.1 so (0.1)^2 = 0.01 < 2
    QED

    your second is again a counterexample one,
    a = 12 b = 4 and c=6
    QED

    -- AI
     
  4. Sep 19, 2004 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    1) A proof is something that must be true for all x and y satisfying those constraints.
    So ypu've not proven the statement. You have found a counter example. I do'nt understand why after finding a counter example you do something else too.
    Your deduction

    "But if x is greater than 1 and y is greater than 1, then y^x is greater than x."

    based upon those two examples is also not true.

    2) do you think it's true? hint that is sometimes used as a definition for what it means fo a to be a prime.
     
  5. Sep 19, 2004 #4
    if u ask me
    a proof is either the one that validates the statement or invalidates it

    so i still accept that as a proof.

    Usuall such questions are tagged with,
    " prove or disprove blah blah blah....."

    -- AI
     
  6. Sep 20, 2004 #5

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    It asked for a proof that the statement is true or a counter example.
    thus we may take the posters use of the word "proof" to indicate proving it true, when they post a counter example instead. (Ie prove it false), but the usage of the word is poor, and confusing, especially given that after giving a counter example, they then "prove" using one example a false statement.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Need help with two simple proofs
Loading...