Course: 2.4 反、反、反、反

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反、反、反、反
Converse, Inverse, and Contrapositive
::反、反、反、反

Consider the statement: If the weather is nice, then I’ll wash the car. We can rewrite this statement using letters to represent the hypothesis and conclusion .
::考虑一下声明:如果天气好的话,我会洗车。我们可以用信件来代表假设和结论,改写声明。

$\begin{aligned} p = the weather is nice & q = I'll wash the car \end{aligned}$
$\begin{align*}p = \text{the weather is nice} && q = \text{I'll wash the car}\end{align*}$
::天气真好,我去洗车

Now the statement is: If $\begin{align*}p\end{align*}$ , then $\begin{align*}q\end{align*}$ , which can also be written as $\begin{align*}p \rightarrow q\end{align*}$ .
::声明是:如果p,那么q,也可以写成pq。

We can also make the negations, or “nots,” of $\begin{align*}p\end{align*}$ and $\begin{align*}q\end{align*}$ . The symbolic version of "not $\begin{align*}p\end{align*}$ " is $\begin{align*}\sim p\end{align*}$ .
::我们也可以对p和q进行否定或“否定”。 “not p”的象征性版本是p。

$\begin{aligned} \sim p = the weather is not nice & \sim q = I won't wash the car \end{aligned}$
$\begin{align*}\sim p = \text{the weather is not nice} && \sim q = \text{I won't wash the car}\end{align*}$
::天气不好,我不洗车

Using these “nots” and switching the order of $\begin{align*}p\end{align*}$ and $\begin{align*}q\end{align*}$ , we can create three new statements.
::使用这些“非”和转换 p 和 q 的顺序,我们可以创建三个新的语句。

$\begin{aligned} Converse & q \to p & \underset{q}{\underset{⏟}{If I wash the car}}, \underset{p}{\underset{⏟}{then the weather is nice}} . \\ Inverse & \sim p \to\sim q & \underset{\sim p}{\underset{⏟}{If the weather is not nice}}, \underset{\sim q}{\underset{⏟}{then I won't wash the car}} . \\ Contrapositive & \sim q \to\sim p & \underset{\sim q}{\underset{⏟}{If I don't wash the car}}, \underset{\sim p}{\underset{⏟}{then the weather is not nice}} . \end{aligned}$
$\begin{align*}& \text{Converse} && q \rightarrow p && \underbrace{\text{If I wash the car}}_{q}, \ \underbrace{\text{then the weather is nice}}_{p}.\\ & \text{Inverse} && \sim p \rightarrow \sim q && \underbrace{\text{If the weather is not nice}}_{\sim p}, \ \underbrace{\text{then I won't wash the car}}_{\sim q}.\\ & \text{Contrapositive} && \sim q \rightarrow \sim p && \underbrace{\text{If I don't wash the car}}_{\sim q}, \ \underbrace{\text{then the weather is not nice}}_{\sim p}.\end{align*}$
::换句话说,如果我洗车,那么天气就很好。如果天气不好,我就不洗车。如果我不洗车,那么天气就不好了。

If the “if-then” statement is true, then the contrapositive is also true. The contrapositive is logically equivalent to the original statement. The converse and inverse may or may not be true. When the original statement and converse are both true then the statement is a biconditional statement . In other words, if $\begin{align*}p \rightarrow q\end{align*}$ is true and $\begin{align*}q \rightarrow p\end{align*}$ is true, then $\begin{align*}p \leftrightarrow q\end{align*}$ (said “ $\begin{align*}p\end{align*}$ if and only if $\begin{align*}q\end{align*}$ ”).
::如果“如果当时”的说法是真实的,那么相反的也是真实的。相反的在逻辑上与原来的说法是等同的。反的和反的可能是或不是真实的。当最初的和反的都是真实的时,该声明就是一种有条件的声明。换句话说,如果pQq是真实的和qQp是真实的,那么pqq(说“如果而且只有q”的话)。

What if you were given a conditional statement like "If I walk to school, then I will be late"? How could you rearrange and/or negate this statement to form new conditional statements?
::如果你们得到一个有条件的声明,比如“如果我走路上学,那么我会迟到?”你们怎么能重新安排和(或)否定这一声明,以形成新的有条件的声明?

Examples
::实例

Example 1
::例1

If $\begin{align*}n > 2\end{align*}$ , then $\begin{align*}n^2 > 4\end{align*}$ .
::如果 n>2, 那么 n2>4 。

Find the converse, inverse, and contrapositive. Determine if each resulting statement is true or false. If it is false, find a counterexample .
::找出相反的、反的和反的。确定每个陈述是真实的还是假的。如果是假的,请找出一个相反的例子。

The original statement is true.
::原始声明是真实的。

$\begin{aligned} \underline{Converse} : & If n^{2} > 4, then n > 2. \\ F a l s e . If n^{2} = 9, n = - 3 or 3. (- 3)^{2} = 9 \\ \underline{Inverse} : & If n \leq 2, then n^{2} \leq 4. \\ F a l s e . If n = - 3, then n^{2} = 9. \\ \underline{Contrapositive} : & If n^{2} \leq 4, then n \leq 2. \\ T r u e . The only n^{2} \leq 4 is 1 or 4 . \sqrt{1} = \pm 1 and \sqrt{4} = \pm 2, which are both less than or equal to 2. \end{aligned}$
$\begin{align*}& \underline{\text{Converse}}: && \text{If} \ n^2 > 4, \ \text{then} \ n > 2. \\ &&& False. \ \text{If} \ n^2 = 9, n = -3 \ \text{or} \ 3. \ (-3)^2=9\\ & \underline{\text{Inverse}}: && \text{If} \ n \le 2,\ \text{then} \ n^2 \le 4. \\ &&& False. \ \text{If} \ n=-3 , \ \text{then} \ n^2=9.\\ & \underline{\text{Contrapositive}}: && \text{If} \ n^2 \le 4, \ \text{then} \ n \le 2. \\ &&& True. \ \text{The only} \ n^2 \le 4 \ \text{is 1 or 4}. \ \sqrt{1}=\pm 1 \ \text{and}\sqrt{4}=\pm 2, \ \text{which are both less than or equal to 2.} \end{align*}$
::反转_ : If n2> 4, 然后 n> 2. false. 如果 n2=9, n%3 或 3. (- 3), 2= 9 反转_ : If n2, 然后 n2+4. False. 如果 n%3, 然后 n2=9. contrac积极性_ : If n2\4, 然后 n2\ 2. TRue. 唯一的 n2+4 是 1 或 4. 1\\\ 1 和 4\\ 2, 它们都小于或等于 2 。

Example 2
::例2

If I am at Disneyland, then I am in California.
::如果我在迪士尼乐园我就在加州

Find the converse, inverse, and contrapositive. Determine if each resulting statement is true or false. If it is false, find a counterexample.
::找出相反的、反的和反的。确定每个陈述是真实的还是假的。如果是假的,请找出一个相反的例子。

The original statement is true.
::原始声明是真实的。

$\begin{aligned} \underline{Converse} : & If I am in California, then I am at Disneyland. \\ F a l s e . I could be in San Francisco. \\ \underline{Inverse} : & If I am not at Disneyland, then I am not in California. \\ F a l s e . Again, I could be in San Francisco. \\ \underline{Contrapositive} : & If I am not in California, then I am not at Disneyland. \\ T r u e . If I am not in the state, I couldn't be at Disneyland. \end{aligned}$
$\begin{align*}& \underline{\text{Converse}}: && \text{If I am in California, then I am at Disneyland.}\\ &&& False. \ \text{I could be in San Francisco.}\\ & \underline{\text{Inverse}}: && \text{If I am not at Disneyland, then I am not in California.}\\ &&& False. \ \text{Again, I could be in San Francisco.}\\ & \underline{\text{Contrapositive}}: && \text{If I am not in California, then I am not at Disneyland.}\\ &&& True. \ \text{If I am not in the state, I couldn't be at Disneyland.}\end{align*}$
::换句话说:如果我在加利福尼亚,那我就在迪士尼乐园,佛尔塞。我可以在旧金山。换句话说:如果我不在迪士尼乐园,那我就不在加利福尼亚。佛尔塞。我也可以在旧金山。联系积极:如果我不在加利福尼亚,那我就不在迪士尼乐园。真的。如果我不在州,我就不能在迪士尼乐园。

Notice for the converse and inverse we can use the same counterexample.
::反反对的注意我们可以使用同样的反比。

Example 3
::例3

Rewrite as a biconditional statement: Any two points are collinear .
::重写为附带条件声明:任何两个点都是共线。

This statement can be rewritten as:
::本语句可改写为:

Two points are on the same line if and only if they are collinear. Replace the “if-then” with “if and only if” in the middle of the statement.
::两点线上是相同的,如果并且只有两点是圆线的。在语句中间,将“如果当时”改为“如果而且只有当时”。

Example 4
::例4

Any two points are collinear.
::任何两个点都是共线的。

Find the converse, inverse, and contrapositive. Determine if each resulting statement is true or false. If it is false, find a counterexample.
::找出相反的、反的和反的。确定每个陈述是真实的还是假的。如果是假的,请找出一个相反的例子。

First, change the statement into an “if-then” statement:
::首先,将声明改为“如果当时”声明:

If two points are on the same line, then they are collinear.
::如果两点在同一条线上,那么它们就是圆线。

$\begin{aligned} \underline{Converse} : & If two points are collinear, then they are on the same line. T r u e . \\ \underline{Inverse} : & If two points are not on the same line, then they are not collinear. T r u e . \\ \underline{Contrapositive} : & If two points are not collinear, then they do not lie on the same line . T r u e . \end{aligned}$
$\begin{align*}& \underline{\text{Converse}}: && \text{If two points are collinear, then they are on the same line.} \ True.\\ & \underline{\text{Inverse}}: && \text{If two points are not on the same line, then they are not collinear.} \ True.\\ & \underline{\text{Contrapositive}}: && \text{If two points are not collinear, then they do not lie on the same line}. \ True.\end{align*}$
::反方向 : 如果两个点是圆线, 那么它们就在同一行上。是的。反方向 : 如果两个点不是在同一行上, 那么它们就不是圆线。正确。 contracive_ : 如果两个点不是圆线, 那么它们就不会在同一行上。正确。

Example 5
::例5

The following is a true statement:
::事实如下:

$\begin{align*}m\angle ABC > 90^\circ\end{align*}$ if and only if $\begin{align*}\angle ABC\end{align*}$ is an obtuse angle .
::mABC>90,如果而且只有@ABC是一个隐蔽角度的话。

Determine the two true statements within this biconditional.
::在此双条件范围内确定两个真实的语句。

Statement 1: If $\begin{align*}m \angle ABC > 90^\circ\end{align*}$ , then $\begin{align*}\angle ABC\end{align*}$ is an obtuse angle.
::报表1:如果 mABC>90,那么ABC是一个隐蔽角度。

Statement 2: If $\begin{align*}\angle ABC\end{align*}$ is an obtuse angle, then $\begin{align*}m\angle ABC > 90^\circ\end{align*}$ .
::报表2:如果ABC是一个隐含角度,则 mABC>90。

Review
::回顾

For questions 1-4, use the statement:
::问题1-4,请使用说明:

If $\begin{align*}AB = 5\end{align*}$ and $\begin{align*}BC = 5\end{align*}$ , then $\begin{align*}B\end{align*}$ is the midpoint of $\begin{align*}\overline{AC}\end{align*}$ .
::如果AB=5和BC=5,那么B是AC的中点。

Is this a true statement? If not, what is a counterexample?
::这是真实的声明吗?如果不是,什么是相反的例子?

Find the converse of this statement. Is it true?
::找到这个说法的反义词,是真的吗?

Find the inverse of this statement. Is it true?
::找到这个说法的反面。

Find the contrapositive of this statement.
::找出这一说法的相反结果。

Find the converse of each true if-then statement. If the converse is true, write the biconditional statement.
::找到每个真实声明的反义词。如果反义词是真实的, 请写下双条件声明。

An acute angle is less than $\begin{align*}90^\circ\end{align*}$ .
::急性角小于90。

If you are at the beach, then you are sun burnt.
::如果你在沙滩上,你就会晒太阳

If $\begin{align*}x > 4\end{align*}$ , then $\begin{align*}x+3>7\end{align*}$ .
::如果 x> 4, 那么 x+3> 7 。

For questions 8-10, determine the two true conditional statements from the given biconditional statements.
::关于问题8-10,确定从特定附带条件的发言中得出的两个真正的有条件声明。

A U.S. citizen can vote if and only if he or she is 18 or more years old.
::美国公民只有在年满18岁或18岁以上时才能投票。

A whole number is prime if and only if its factors are 1 and itself.
::整个数字是首要的,如果而且只有当其因素是1和本身时。

$\begin{align*}2x = 18\end{align*}$ if and only if $\begin{align*}x = 9\end{align*}$ .
::2x=18 如果且只有 x=9 时为2x=18。

Review (Answers)
::回顾(答复)

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。

Resources
::资源

Section outline

General

反、反、反、反

Converse, Inverse, and Contrapositive ::反、反、反、反

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Converse, Inverse, and Contrapositive
::反、反、反、反

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源