True or false? $x^2\ne x\implies x\ne 1$

Today I had an argument with my mathematics educator at college. We were addressing some straightforward True/False inquiries and also among the inquiries was the following:

$$x^2\ne x\implies x\ne 1$$

I quickly addressed real, but also for some factor, every person (including my schoolmates and also mathematics educator) is differing with me. According to them, when $x^2$ is not equivalent to $x$ , $x$ additionally can not be $0$ and also due to the fact that $0$ isn't left out as a feasible value of $x$ , the sentence is incorrect. After hrs, I am still incapable to recognize this extremely straightforward effects. I can not think I'm stuck to something so simple.¢¢ Why I assume the sensible sentence over is true: ¢ My understanding of the effects icon $\implies$ is the following: If the left component holds true, after that the appropriate component has to be additionally real. If the left component is incorrect, after that absolutely nothing is claimed concerning the appropriate component. In the appropriate component of this details effects absolutely nothing is claimed concerning whether $x$ can be $0$ . Possibly $x$ can not be $-\pi i$ also, yet as I see it, it does not actually issue, as long as $x \ne 1$ holds. And also it constantly holds when $x^2 \ne x$ , consequently the sentence holds true.

TL ;DR:

$x^2 \ne x \implies x \ne 1$ : Is this sentence real or incorrect, and also why?

Sorry for troubling such an impressive area with such a straightforward inquiry, yet I needed to ask a person.

103
2022-06-30 03:20:05
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Answers: 2

The short answer is: Yes, it is true, because the contrapositive just expresses the fact that $1^2=1$.

But in controversial discussions of these issues, it is often (but not always) a good idea to try out non-mathematical examples:


"If a nuclear bomb drops on the school building, you die."

"Hey, but you die, too."

"That doesn't help you much, though, so it is still true that you die."


"Oh no, if the supermarket is not open, I cannot buy chocolate chips cookies."

"You cannot buy milk and bread, either!"

"Yes, but I prefer to concentrate on the major consequences."


"If you sign this contract, you get a free pen."

"Hey, you didn't tell me that you get all my money."

"You didn't ask."


Non-mathematical examples also explain the psychology behind your teacher's and classmates' thinking. In real-life, the choice of consequences is usually a loaded message and can amount to a lie by omission. So, there is this lingering suspicion that the original statement suppresses information on 0 on purpose.

I suggest that you learn about some nonintuitive probability results and make bets with your teacher.

79
2022-07-01 15:24:03
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First, some general remarks about logical implications/conditional statements.

  1. As you know, $P \rightarrow Q$ is true when $P$ is false, or when $Q$ is true.

  2. As mentioned in the comments, the contrapositive of the implication $P \rightarrow Q$, written $\lnot Q \rightarrow \lnot P$, is logically equivalent to the implication.

  3. It is possible to write implications with merely the "or" operator. Namely, $P \rightarrow Q$ is equivalent to $\lnot P\text{ or }Q$, or in symbols, $\lnot P\lor Q$.

Now we can look at your specific case, using the above approaches.

  1. If $P$ is false, ie if $x^2 \neq x$ is false (so $x^2 = x$ ), then the statement is true, so we assume that $P$ is true. So, as a statement, $x^2 = x$ is false. Your teacher and classmates are rightly convinced that $x^2 = x$ is equivalent to ($x = 1$ or $x =0\;$), and we will use this here. If $P$ is true, then ($x=1\text{ or }x =0\;$) is false. In other words, ($x=1$) AND ($x=0\;$) are both false. I.e., ($x \neq 1$) and ($x \neq 0\;$) are true. I.e., if $P$, then $Q$.
  2. The contrapositive is $x = 1 \rightarrow x^2 = x$. True.
  3. We use the "sufficiency of or" to write our conditional as: $$\lnot(x^2 \neq x)\lor x \neq 1\;.$$ That is, $x^2 = x$ or $x \neq 1$, which is $$(x = 1\text{ or }x =0)\text{ or }x \neq 1,$$ which is $$(x = 1\text{ or }x \neq 1)\text{ or }x = 0\;,$$ which is $$(\text{TRUE})\text{ or }x = 0\;,$$ which is true.
57
2022-07-01 15:22:49
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