TITLE: True or false? $x^2\ne x\implies x\ne 1$
QUESTION [101 upvotes]: Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:
$$x^2\ne x\implies x\ne 1$$
I immediately answered true, but for some reason, everyone (including my classmates and math teacher) is disagreeing with me. According to them, when $x^2$ is not equal to $x$, $x$ also can't be $0$ and because $0$ isn't excluded as a possible value of $x$, the sentence is false. After hours, I am still unable to understand this ridiculously simple implication. I can't believe I'm stuck with something so simple.
Why I think the logical sentence above is true:
My understanding of the implication symbol $\implies$ is the following:
If the left part is true, then the right part must be also true. If the left part is false, then nothing is said about the right part. In the right part of this specific implication nothing is said about whether $x$ can be $0$. Maybe $x$ can't be $-\pi i$ too, but as I see it, it doesn't really matter, as long as $x \ne 1$ holds. And it always holds when $x^2 \ne x$, therefore the sentence is true.
TL;DR:
$x^2 \ne x \implies x \ne 1$: Is this sentence true or false, and why?
Sorry for bothering such an amazing community with such a simple question, but I had to ask someone.

REPLY [2 votes]: Introduction
You asked whether the following statement is true or not:

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

Another way to write the above is as:

if $(x^2 \ne x)$ then $(x \ne 1)$

The arrow symbol ($\implies$) is similar, but not quite the same, as the English phrase if....then....
The contra-positive of $(P \implies Q)$
Okay now: suppose that you have the statement of the form "If $P$ then $Q$"
$P$ and $Q$ can be any true or false statements of your choice.
For example, $P$ could be the statement "pillows are soft"
"If $P$ then $Q$" is also sometimes written as $(P \implies Q)$
There is a well known theorem in logic which states the following:

For any $P$ and $Q$ taken from the set $\{$ true, false $\}$, the following is true:
$(\text{not } P \implies \text{not } Q)$ if and only if $(Q \implies P)$

That means that the following two statements are equivalent:

if $(x^2 \ne x)$ then $(x \ne 1)$
if $(x = 1)$  then $(x^2 = x)$

In some sense, the statement says that $1^2 = 1$, which is true enough.
Sets
It considered to be very bad math to write things like the following:

Is it true or false that $(x^2 \ne x \implies x \ne 1)$?
Find $x$ such that $3 + (x-5)^{2} = 0$

This is because the above examples fail to explain what $x$ is.

Is $x$ a whole number, such as $1, 2, 3, \dots, 98, 99, 100$?
Is $x$ a decimal number, such as $\sqrt{2}$
Is $x$ a non-real complex number, such as $20 + 5*i$?

You are supposed to write things like this instead:

Is the following true or false?

For every real number $x, (x^2 \ne x \implies x \ne 1)$

Find every complex non-real number $x$ such that:

$3 + (x-5)^{2} = 0$


My last example uses the non-real number $i$
$i*i = -1$
I think if you think it will help if you start using "sets."
The following is an example of a set:

my_set $= \{1, 3, 6, 7, 22\}$

A set is like a suitcase full of clothing
A set is also like a cookie jar, or cardboard box.
A set is a container.
A suitcase might contain a t-shirt.
Well, my_set contains the numbers $1, 3, 6, 7,$ and $22$.

The number $3$ is like a t-shirt in the sense that the number $3$ is inside the suitcase.
I went to public school in the United States.
I did not see sets until I was in college.
Sets are basic, basic math.
sets are more basic than knowing how to compute $4.5/0.3$
You are not allowed to write math like $x^2 \ne x \implies x \ne 1$ unless you first tell the reader what set $x$ comes from.
The following is a very ugly formula for a function named $WEIRD$:

$\text{WEIRD_FUNC}(X) = [1-w(x)]*\text{LEFT_PIECE}(X) + [w(x)]*\text{RIGHT_PIECE}(X)$
$\text{LEFT_PIECE}(X) = (x+10)*(x+5)$
$\text{RIGHT_PIECE}(X) = 3 + (x-7)^{2}$
$W(x) = \frac{tanh(10)}{2} + \frac{tanh(x)}{2}$

A plot of the weird function is shown below:

Suppose I asked you,

"Find all $x$ such that $WEIRD(x) = 0$"

The answers vary depending on which set $x$ is taken from.
The set of all integers $x$ such that $WEIRD(X) = 0$ is $\{-10\}$
The set of all real numbers $x$ such that $WEIRD(X) = 0$ is approximately $\{-10, -5.01\}$
The set of all complex numbers $x$ such that $WEIRD(X) = 0$ is approximately $\{-10, -5.01, 7+i*\sqrt{3}, 7-i*\sqrt{3}\}$
Analyzing your classmates argument regarding the number zero
Is the following statement true or not?

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

The following is your description of your teacher, and/or classmates, reasoning:


When $x^2$ is not equal to $x$, $x$ also can't be $0$.
$0$ is not excluded as a possible value of $x$
some sentence, or another, is false.


That description is very difficult to understand.
I will say that the following three statements are equivalent to each-other.
Also, all statements are false:

$\forall x \in \mathbb{R}, x^2 \ne x$
for any decimal number $x$, $x^2 \ne x$
if $x$ is a real number, then $x^2 \ne x$

Here is a proof:




Line No.
statement
justification




0
$0$ exists
axiom


1
$0$ is a decimal number.
axiom


2
$0^{2} = 0$
axiom


3
not $(0^2 \neq 0)$
from line 2


4
there exists a decimal number $x$ such that not $(x^2 \neq x)$
from lines 0,1,3


5
not for every decimal number $x$, $(x^2 \neq x)$
from line 4




Notice that your teacher said, "possible value of $x$"
In mathematics, the symbol ◇ is sometimes used as short-hand notation for the word "possible"
All of the following are logically equivalent:


it is not possible for me to see a movie this weekend
$NOT$ ◇ for me to see a movie this weekend
It is necessary for me to NOT see a movie this weekend
▢ for me to NOT see a movie this weekend


The statement "$0$ is not excluded as a possible value of $x$" can be written as:


not(not ◇ $x = 0$)
It is not the case that it is not possible that $x$ is zero.


The above is equivalent to "it is possible that $x = 0$"
I think that I can write a proof outline of what your professor and/or classmates were trying to say




Line No.
statement
justification




$0$
◇ $x = 0$
axiom


$1$
▢$(x^2 \ne x) \implies$ not ◇ $x = 0$
axiom


$2$
▢ $x^2 \ne x$
axiom


$3$
not ◇ $x = 0$
1,2 modus ponens


4
◇ $x = 0$ and not ◇ $x = 0$
lines 0, 3 conjunction


5
$NOT$ ▢$(x^2 \ne x)$
from line 4, reject 2




I think that your teacher was saying that if $x = 0$ is possible, then it is not necessary that $x^2 \ne x$
$0 \in S \implies NOT(\forall x \in S, x^2 \ne x)$
One last Note
All of the following are equivalent:


(it is necessary that $x^2 \neq x$) implies that (it is necessary that $x \neq 1$)
(it is NOT necessary that $x \neq 1$) implies that (it is NOT necessary that $x^2 \neq x$)
(it is possible that $x = 1$) implies that (it is possible that $x^2 = x$)
For any set $S$, [(there exists $x$ in $S$ such that $x = 1$) implies that (there exists $x$ in $S$ such that $x^2 = x$)]