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
- 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.
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.
- 7 年前
this is mathematics and computer science question
and need to prove it is true or disprove it is fales
- MingLam ChoiLv 77 年前