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Eason 發問時間: 社會與文化語言 · 7 年前

Question from my assignment

Consider the following proposition, where the domain of denition is the set of RATIONALS:

(universal quantifier)m (universal quantifier)n (existential quantifier)t (m < t < n):

If it is true, prove it, if it is false, disprove it.

i can't typing symbols for

(universal quantifier) and (existential quantifier)

plz answer my question as soon as possible XD

3 個解答

評分
  • 7 年前
    最佳解答

    The statement is true.

    Claim: Between any two distinct rational numbers m and n where m<n, there

    exists a rational number t such that m<t<n.

    Proof:

    Consider t = (m+n)/2

    1) Clearly, m<t<n; this is obvious

    2) Show that t is rational.

    Recall that any rational number can be expressed as a fraction of integers.

    Let m = a/b and n = c/d, for a,b,c,d \in Z

    Then t = (ad+bc)/2bd which is also rational because ad+bc and 2bd are integers.

    *For this part, alternately you can just recall that rationals are closed under

    addition, subtraction, multiplication, as well as division by a nonzero rational.

    Hence we have constructed a rational number t such that m<t<n, for any two

    distinct rational numbers m, n (m<n). Q.E.D.

    P.S. In fact, there are infinitely many other rational numbers between two distinct rational numbers.

    I hope this helps. Let me know if you have questions.

    參考資料: Myself.
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  • 7 年前

    this is mathematics and computer science question

    and need to prove it is true or disprove it is fales

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  • 7 年前

    你要問什麼?

    這是英文題還是數學題?

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