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X^2<x, then x<1

  1. Dec 2, 2008 #1
    1. The problem statement, all variables and given/known data

    x^2<x, then x<1

    2. Relevant equations

    3. The attempt at a solution
    We will use a proof by contrapositive.
    We assume x>1 and we want to show x^2>x.
    Let x be greater than 1 and let x^2=x*x.
    If x^2=x*x, then x*x>x. Therefore, x^2>x.
  2. jcsd
  3. Dec 2, 2008 #2
    Not quite, your contrapositive is correct and your first step assuming x > 1 is correct. But then you said:

    Firstly x^2 = x*x is just taken for granted so you don't need to put it in. Secondly, why does x^2=x*x imply x*x > x ?

    All we know is:
    x > 1
    Then multiplying both sides by x we get:
    x*x > 1*x
    so then
    x^2 > x

    We can also prove this without the contrapositive.
    Last edited: Dec 2, 2008
  4. Dec 2, 2008 #3
    So, is this all I would have to do - the multiplying by x?
  5. Dec 2, 2008 #4
    Think about it for a little while. Every step has to follow logically from the previous and all we had to work with was a single assumption. Search the forum for similar inequality proofs or do a Google search on introduction to proofs. If you have a textbook with a section on proofs and the different kinds, that would also help looking at.
  6. Dec 2, 2008 #5


    Staff: Mentor

    As kidmode pointed out, you can do this directly. You also need to assume that x > 0, because the inequality isn't true otherwise. E.g., if x = -1/2, x^2 = 1/4 > -1/2, and if x = -2, x^2 = 4 > -2.

    Start by assuming x > 0 and that x^2 < x.
    Then x^2 - x < 0.
    Now factor the left side and determine which values of x make it true.
  7. Dec 2, 2008 #6
    Whoops. I wrote it down but I didn't add it, thanks Mark44.
  8. Dec 3, 2008 #7


    User Avatar
    Staff Emeritus
    Science Advisor

    I hate to add a complication but the negation of "x< 1" is NOT "x> 1".
  9. Dec 3, 2008 #8
    Whoops again.

    ~(x<1) = (x >= 1)

    Just put a line underneath all the inequalities in your contrapositive proof lol.
    You'll have to fix the negation of the right side of "if" statement.
    You'll end up proving:

    x^2 >= x

    Thanks HallsofIvy
  10. Dec 3, 2008 #9
    Well, i'm supposed to prove using the contrapositive.
  11. Dec 3, 2008 #10
    "P implies Q" and "Not Q implies Not P" are logically equivalent. Constructing a truth table will show that the truth values of both statements are the same. So proving the statement is true or it's contrapositive is true are equivalent.
  12. Dec 3, 2008 #11


    Staff: Mentor

    That wasn't clear to me from your first post "We will use a proof by contrapositive."

    I interpreted that to mean that this was the direction you had decided to go, not one that was mandated in the problem.
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