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One-to-one function and range

  1. Sep 18, 2009 #1
    1. The problem statement, all variables and given/known data

    f : R+ → R+ defined by f(x) = 2x
    h : R+ → N defined by h(x) = ⌊x⌋ (the largest integer ≤ x)
    g : 2ᴿ → 2ᴿ defined by g(A) = A ∪ [0,1]

    2. Relevant equations


    3. The attempt at a solution

    I know that every element of the domain has to correspond to exactly one element on the co-domain {range?}. But not sure how to determine if it satisfies the condition of one-to-one and its range.
     
  2. jcsd
  3. Sep 18, 2009 #2

    Office_Shredder

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    Ok, that's the definition of a function. The definition of one-to-one is that every element in the range corresponds to exactly one element in the domain. For example, in number 2. Is there more than one x such that h(x)=1?
     
  4. Sep 18, 2009 #3
    I would say that there is only one which is x=1. If I always plug the same number to the x1 and x2, the answer seems to be always same. for example,

    if h(x₁) = h(x₂), then ⌊x₁⌋ = ⌊x₂⌋, so x₁ = x₂ This is why I'm confused.

    Am I correct?
     
  5. Sep 18, 2009 #4

    Office_Shredder

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    ⌊1.2⌋=⌊1.1⌋?
     
  6. Sep 18, 2009 #5
    no, ⌊1.2⌋ is not equal to ⌊1.1⌋, but if so it means ⌊x₁⌋ is not equal to ⌊x₂⌋. In definition, the function if injective if a≠ b, then f(a)≠ f(b). So, is it one-to-one function?
     
  7. Sep 18, 2009 #6

    Office_Shredder

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    I'm confused. What are ⌊1.1⌋ and ⌊1.2⌋ then? I would have though they're both 1
     
  8. Sep 18, 2009 #7
    What is the greatest integer less than or equal to 1.2?
    What is the greatest integer less than or equal to 1.1?

    These are unequal?

    --Elucidus
     
  9. Sep 18, 2009 #8
    oh~ok. I got that now. If so, in this case, ⌊x₁⌋ ≠ ⌊x₂⌋ but h(x₁) = h(x₂). I can say it is not one-to-one function. Correct?
    I also have to say its range, how can I put?

    And also, I have a problem to define A ∪ [0,1]. I know [0,1] means the set [xɛR | 0 ≤x ≤1]. But then, I'm not sure what I have to do..
     
  10. Sep 19, 2009 #9

    Office_Shredder

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    h(x₁) = h(x₂)

    this is equivalent to
    ⌊x₁⌋ = ⌊x₂⌋

    Since h(x) = ⌊x⌋

    I think you need to review the definition of greatest integer less than or equal to.

    The same thing for the third part... it looks like you're unclear with concepts that your class is assuming you're familiar with. I highly recommend that you find out when you can go see your teacher for out of class help to catch you up to speed with the rest of the class
     
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