Math Word Problem

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Let A be the set containing all rational numbers that are less than 5. Is there a rational number q in set A such that all other numbers in set A are less than q? Why or why not?

  • Math Word Problem -

    We will try to prove this by contradiction.

    Hypothesis: existence of q=a/b which is the largest rational number less than 5, i.e. q=a/b<5 and b≠0, and that no other rational number exists that is larger than q and less than 5.

    We will calculate the r, the average between q and 5
    r=(q+5)/2
    =(a/b+5)/2
    =(a+10b)/2b
    so that
    q<r<5
    which means that
    r is greater than q,
    r is less than 5, and
    r is rational.

    Therefore that hypothesis that q exists is false.

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