2.5 预测和反示例

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  假设和对应示例

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  Conjectures and Counterexamples
  ::假设和对应示例

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  A conjecture is an “educated guess” that is based on examples in a pattern. A counterexample is an example that disproves a conjecture.
  ::一种假设是一种基于模式实例的“教育猜想 ” 。 反例就是否定一种猜想的例子。

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  Suppose you were given a mathematical pattern like h = $\begin{array}{r}-16/t^2\end{array}$ . What if you wanted to make an educated guess, or conjecture, about h ?
  ::假设给了你像 h = ~ 16/ t2 这样的数学模式,如果你想做一个教育性的猜测,或者推测,大约是 h?

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  Examples
  ::实例

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  Use the following information for Examples 1 and 2:
  ::实例1和2使用以下信息:

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  A car salesman sold 5 used cars to five different couples. He noticed that each couple was under 30 years old. The following day, he sold a new, luxury car to a couple in their 60’s. The salesman determined that only younger couples by used cars.
  ::一个汽车销售员向五对不同的夫妇出售了五辆旧车。 他注意到每对夫妇都不到30岁。 第二天,他向60年代的一对夫妇出售了一辆新的豪华轿车。 销售员确定只有年轻夫妇才用二手车。

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  Example 1
  ::例1

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  Is the salesman’s conjecture logical? Why or why not?
  ::销售员的推测合乎逻辑吗? 为什么或为什么?

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  It is logical based on his experiences, but is not true.
  ::这是根据他的经验合乎逻辑的,但事实并非如此。

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  Example 2
  ::例2

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  Can you think of a counterexample?
  ::你能想到一个反面的例子吗?

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  A counterexample would be a couple that is 30 years old or older buying a used car.
  ::30岁或30岁以上的一对夫妇购买一辆二手车,将是一个反实例。

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  Example 3
  ::例3

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  Here’s an algebraic equation and a table of values for $\begin{array}{r}n\end{array}$ and $\begin{array}{r}t\end{array}$ .
  ::这是代数方程式和n和t的数值表。

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  $$\begin{array}{r}t=(n-1)(n-2)(n-3)\end{array}$$

  
  ::t=(n-1)(n-2)(n-3)

  $\begin{array}{r}n\end{array}$	$\begin{array}{r}(n-1)(n-2)(n-3)\end{array}$	$\begin{array}{r}t\end{array}$
  1	$\begin{array}{r}(0)(-1)(-2)\end{array}$	0
  2	$\begin{array}{r}(1)(0)(-1)\end{array}$	0
  3	$\begin{array}{r}(2)(1)(0)\end{array}$	0

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  After looking at the table, Pablo makes this conjecture:
  ::Pablo在查看桌子后,

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  The value of $\begin{array}{r}(n-1)(n-2)(n-3)\end{array}$ is 0 for any number $\begin{array}{r}n\end{array}$ .
  :n-1) (n-2) (n-3) 值为 0, 任何 n 数值为 0 。

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  Is this a true conjecture?
  ::这是真的猜测吗?

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  This is not a valid conjecture. If Pablo were to continue the table to $\begin{array}{r}n=4\end{array}$ , he would have see that $\begin{array}{r}(n-1)(n-2)(n-3)=(4-1)(4-2)(4-3)=(3)(2)(1)=6\end{array}$
  ::这不是一个有效的假设。 如果巴勃罗继续将表格的n=4, 他就会看到(n-1)(n-2)(n-3)=(4-1)(4-2)(4-3)=(3)(2)(1)=6)

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  In this example $\begin{array}{r}n=4\end{array}$ is the counterexample.
  ::在此示例中, n=4 是反例 。

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  Example 4
  ::例4

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  Arthur is making figures for an art project. He drew polygons and some of their diagonals .
  ::Arthur正在为艺术项目做数字,他画了多边形和一些对角形。

  

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  From these examples, Arthur made this conjecture:
  ::Arthur从这些例子中得出这样的推测:

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  If a convex polygon has $\begin{array}{r}n\end{array}$ sides, then there are $\begin{array}{r}n-2\end{array}$ triangles formed when diagonals are drawn from any vertex of the polygon.
  ::如果锥形多边形有正边,则在从多边形的任何顶端抽取对角时形成n-2三角形。

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  Is Arthur’s conjecture correct? Or, can you find a counterexample?
  ::亚瑟的猜想是否正确? 或者,你能找到相反的例子吗?

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  The conjecture appears to be correct. If Arthur draws other polygons, in every case he will be able to draw $\begin{array}{r}n-2\end{array}$ triangles if the polygon has $\begin{array}{r}n\end{array}$ sides.
  ::猜想似乎是正确的。 如果亚瑟绘制其他多边形, 在每种情况下, 如果多边形有 n- 2 边, 他就可以绘制 n-2 三角形 。

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  Notice that we have not proved Arthur’s conjecture, but only found several examples that hold true. So, at this point , we say that the conjecture is true.
  ::我们注意到我们没有证明阿瑟的推测,但只发现了几个符合事实的例子。 因此,在这一点上,我们说推测是真实的。

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  Example 5
  ::例5

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  Give a counterexample to this statement: Every prime number is an odd number.
  ::给此语句一个反例: 每个质数是一个奇数 。

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  The only counterexample is the number 2: an even number (not odd) that is prime.
  ::唯一的反例是数字2:一个偶数(非奇数),这个数字是质数。

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  Review
  ::回顾

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  Give a counterexample for each of the following statements.
  ::对以下各语句进行反比。

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  1. If $\begin{array}{r}n\end{array}$ is a whole number, then $\begin{array}{r}{n}^2>n\end{array}$ .
     ::如果 n 是一个整数,则 n2>n。
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  2. All numbers that end in 1 are prime numbers.
     ::以1结尾的所有数字都是质数。
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  3. All positive fractions are between 0 and 1.
     ::所有正分数在0到1之间。
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  4. Any three points that are coplanar are also collinear.
     ::任何三个点的共平面点也是共线点。
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  5. All girls like ice cream.
     ::所有女孩都喜欢冰淇淋
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  6. All high school students are in choir.
     ::所有高中学生都参加合唱团。
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  7. For any angle there exists a complementary angle.
     ::对于任何角度都存在一个互补角度。
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  8. All teenagers can drive.
     ::所有青少年都能开车
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  9. If $\begin{array}{r}n\end{array}$ is an integer, then $\begin{array}{r}n>0\end{array}$ .
     ::如果 n 是整数,则 n>0。
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  10. All equations have integer solutions.
      ::所有方程式都有整数解决方案 。

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源