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Homework Help: Discrete math & sentences

  1. Jan 25, 2008 #1
    Rewrite the following statement formally. Use variables and include both quantifiers [tex]\forall[/tex] and [tex]\exists[/tex] in your answer.

    Statement: Every rational number can be written as a ratio of some two integers.

    If I didn't have to use [tex]\exists[/tex] I'd write it as follows

    [tex]\forall[/tex]rational numbers x, x is a ratio of two integers.

    But I can't think of a way or any reason why I'd want to include the quantifier [tex]\exists[/tex].
  2. jcsd
  3. Jan 25, 2008 #2
    You must use variables in your answer.
  4. Jan 26, 2008 #3


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    What does "x is a ratio of two integers" mean? That's where you need [itex]\exists[/itex].
  5. Jan 26, 2008 #4
    Ok thanks, I rewrote it as
    [tex]\forall[/tex] rational numbers x, [tex]\exists[/tex] a rational number y and a rational number z such that x=y/z.

    One more question if anyone has time to help me with.

    Rewrite the statement formally.
    Statement: There is a program that gives the correct answer to every question that is posed to it.

    So I rewrote it as
    [tex]\exists[/tex] a program p such that [tex]\forall[/tex]questions q, p always answers q correctly.
    Is this incorrectly because I have "correctly" as the final word?
    The book shows the answer as
    [tex]\exists[/tex] a program P such that [tex]\forall[/tex] questions Q posed to P, P gives the correct answer to Q.

    I didn't know if these was some technicality that would make my answer incorrect as apposed to the book's answer.
  6. Jan 26, 2008 #5
    It's mostly right except the original statement is that any rational number can be written as a ratio of 2 integers.

    For the second one, the only real problem I see is that "always" is redundant.
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