What do we mean when we say that x∈R

[SHASHWAT PRATAP SING]

When we say that x∈R does here x represents all real numbers at the same time or x represents only a single real number at one time.
How can a single variable(x) represent all the real numbers at the same time ?
when we write x∈R it means that x is an element of the set of real numbers, this means that x represents a single real number but then why we start to treat it as if x represents all the real numbers at once as in inequality suppose we have x>-2 this means that x can be any real number greater than -2 but then why we say that all the real numbers greater than -2 are the solutions of the inequality. x should represent a single real number right ? then how can x represents all the real numbers at the same time.Please help me...

 

[etotheipi]

To let $x \in \mathbb{R}$ and $x>-2$ means to let that $x$ is an arbitrary element of the set $\mathbb{R} - (-\infty, -2]$, which is a subset of $\mathbb{R}$.

 

[BvU]

$\mathbb R$ is a collection of infinite size. $x\in \mathbb R$ says something about $x$, namely that it is part of the collection $\mathbb R$.

why we start to treat it as if x represents all the real numbers at once

We don't. Based on $x\in \mathbb R$ alone, we don't know anything else about $x$. It may not even exist.

x should represent a single real number right ?

Not necessarily:

$x\in \mathbb R\ \ \& \ \ x^2 = -1\qquad$ no solution.​

$x\in \mathbb R \ \ \& \ \ x^2 = \ \ 1\qquad$ two solutions.​

$x\in \mathbb R \ \ \& \ \ x>1 \ \ \&\ \ x < 1.0000000000000001\qquad$ inifnite number of solutions.​

 

[Gaussian97]

I think the answer is that $x$ represent a single number. If I say $x\in \mathbb{N}$, are I'm saying that $x$ is 1? No, of course not, but I'm saying that it could be 1. At the same time, I'm not saying it is 2, but that it could be 2, but in any case, I'm saying that $x$ is equal to 1 and 2 simultaneously. Of course what I'm saying is that, for example, $x$ cannot be $3/2$.

So, when we say that $x$ belongs to a set, we don't mean that $x$ is all the elements of the set, but just that it could be any element, we just don't know which one (and usually we don't care which one is).
Let's look at the examples BvU said:
$$x\in \mathbb{R}\ \land\ x^2 = -1$$
we are saying that $x$ could be ANY real number that also has the property $x^2 = -1$. Of course, no such real number exists... So $x$ cannot be any number at all. In the example
$$x\in \mathbb{R}\ \land\ x^2 = 1$$
we are saying that $x$ could be ANY real number that also has the property $x^2 = 1$. As BvU said there are two such numbers, i.e. 1 and -1. Does that mean that $x$ is both numbers simultaneously? Not, of course not. Just mean that it is one of the two numbers, we just don't know (don't care) which one. Is like if I tell you that I'm thinking of a number, that can be 1 or -1. You just don't know which number I'm thinking about. And the same is true for the example
$$x\in \mathbb{R}\ \land\ x > 1\ \land \ x< 1.01$$
there are infinitely many numbers satisfying these conditions, but $x$ is just one of those, not all those.

 

[fresh_42]

When we say that x∈R does here x represents all real numbers at the same time or x represents only a single real number at one time.

The existence quantifier is usually noted explicitly: ... for some $x\in \mathbb{R}$..., ...there is an $x\in \mathbb{R}$..., or the point is specified by an index: ... $x_0\in \mathbb{R}$ ... which is always a certain point.

If none of this is said, and we read: ...let $x\in \mathbb{R}$ ... with or without restriction ...such that ..., or ... given $x\in \mathbb{R}$ ..., or simply $(x\in \mathbb{R})$ then it is normally an all quantifier, meaning any $x\in \mathbb{R}$ possibly restricted to a subset. All here means any, i.e. we name it $x\in \mathbb{R}$ so we can work with it, but as long as it isn't restricted by following arguments, such an $x\in \mathbb{R}$ is an arbitrary one. If it remains arbitrary, then the statement holds for all $x\in \mathbb{R}$.

The exact notation would be $\exists \;x\in \mathbb{R}$ or $\forall \;x\in \mathbb{R}$.

 

[PeroK]

When we say that x∈R does here x represents all real numbers at the same time or x represents only a single real number at one time.
How can a single variable(x) represent all the real numbers at the same time ?
when we write x∈R it means that x is an element of the set of real numbers, this means that x represents a single real number but then why we start to treat it as if x represents all the real numbers at once as in inequality suppose we have x>-2 this means that x can be any real number greater than -2 but then why we say that all the real numbers greater than -2 are the solutions of the inequality. x should represent a single real number right ? then how can x represents all the real numbers at the same time.Please help me...

Another way to look at this is to think of the properties that the real numbers have. If you let $x \in \mathbb R$, then $x$ is something with the properties that all real numbers have. If you then have $x > 0$, then $x$ is something that has the properties that all positive real numbers have. And so on.

It may help to find a real world analogy. You might say let $A$ be a person. Whatever you say about $A$ then must have only the properties that all people share. If you then say that $A$ is male, then $A$ must have only those properties that all men share etc. With real life, of course, even gender gets complicated!

The point is that you do not need to have chosen a particular element or person. All you are assuming is that the element or person you are dealing with has certain properties or characteristics. And you proceed by logical deduction using on those properties.

 

[SHASHWAT PRATAP SING]

okay I understood that when we write x∈R this means x is an element of the set R,x represents just a SINGLE real number , Since we have not specified exactly which element x represents that's why we say x can represent any real number.
But, then since, x represents only a single real number which is unspecified then why do we use statements like "for all x in a set" what does this mean ?
Also, for the inequality x>2 If x represents just a SINGLE real number which can be any real number greater than -2,then why do we say that x>2 inequality has infinite solutions ? as it just has a single solution that is x.
Please Help me...

 

[BvU]

I understood that when we write x∈R this means x is an element of the set R

Period.

x represents just a SINGLE real number

No it does not say that. You misunderstood. It does not even say that $x$ exists, whether it is one single number or a finite or infinite set of numbers. That information should come from other statements.

 

[pbuk]

why do we use statements like "for all x in a set"

Because we are careless and because the symbol $\forall$ reminds us of words starting with 'A' like 'all'. We should instead say "for each x in..." or "for every x in...", because we mean that the statement that follows is true for each element of the set taken one at a time, not all at once.

Also, for the inequality x>2 If x represents just a SINGLE real number which can be any real number greater than -2,then why do we say that x>2 inequality has infinite solutions ? as it just has a single solution that is x.

Do you really think that there is only one solution to $x > 2$? Of course not, clearly there are infinitely many solutions. Why are you trying to build an interpretation that does not make sense?

 

[SHASHWAT PRATAP SING]

I want to understand the full concept of the variable x,
what does it mean when we say x∈R ? I want understand the concept in detail.
if x is an element of the set R, then how can every element be x it is just like saying- {x,x,x,x,x,x,x}. That's why I am asking what does "for each x in ..." or "for every x in..." mean.
I want to understand this Set and variable x concept from scratch...
Help me...

 

[Gaussian97]

Let me start by saying that I'm not a mathematician and I haven't studied sets in a complete formal and rigorous way, so probably what I'll say now is wrong and I have misunderstood lots of things. But just to share my point of view let me answer as how I see this:

No it does not say that. You misunderstood. It does not even say that $x$ exists, whether it is one single number or a finite or infinite set of numbers. That information should come from other statements.

When we say $x \in S$ where $S$ is a set, $x$ indeed represent a single element of the set $S$ and, unless $S$ is the empty set, it will always exist. When I say $x \in \mathbb{R}$, $x$ is a single (real) number, not the whole set of real numbers, if the second were correct we would have $x \in \mathbb{R} \Longrightarrow x = \mathbb{R}$ and the symbol $\in$ would be meaningless.

okay I understood that when we write x∈R this means x is an element of the set R,x represents just a SINGLE real number , Since we have not specified exactly which element x represents that's why we say x can represent any real number.
But, then since, x represents only a single real number which is unspecified then why do we use statements like "for all x in a set" what does this mean ?
Also, for the inequality x>2 If x represents just a SINGLE real number which can be any real number greater than -2,then why do we say that x>2 inequality has infinite solutions ? as it just has a single solution that is x.
Please Help me...

Well, this can be rather abstract maybe, but let me do an example. I could say
$$x \in \mathbb{R}, \qquad x + 2 > 0$$
here $x$ can be any real number, is not all the real numbers, is just a single real number, but since we don't care which one, you can choose anyone. And then $x + 2 > 0$ is a well-defined statement. This is completely different of saying that $x + 2 > 0$ is True. Of course not for all real number the statement is True, and that's why we need Quantifiers to be able to say things like:
$$\forall x \in \mathbb{R}, \qquad x^2 \geq 0 \text{ is True}$$
Or, if $f: \mathbb{R} \to \mathbb{R}$ is a continuous function
$$\lim_{x \to \pm \infty} f(x) = \pm \infty \Longrightarrow \exists x \in \mathbb{R} \quad f(x)=0 \text{ is True}$$
In this case $x$ can be any real number, and for any real number $x$ the statement $f(x)=0$ is well-defined. But only for some of them is True.

Let's return to the example of $x \in \mathbb{R}, \ x + 2 > 0$ you could ask ¿Why we even need the $x \in \mathbb{R}$? ¿Is not obvious that $x + 2 > 0$ will be True or False? And the answer is no, the $x\in \mathbb{R}$ is essential. If I change the $x \in \mathbb{R}$ for $x \in \mathbb{C}$ the statement $x + 2 > 0$ has no sense and is not True of False, because complex numbers have no order. The same is true if we write $x \in S$ for some arbitrary set, for example, the set of all the animals. What the heck would mean to add 2 to an animal?

I hope this will help you to understand what $x \in \mathbb{R}$ means, and why is different to say $x \in \mathbb{R}$ and $\forall x \in \mathbb{R}$.

Now it's time to repeat me and remember that I'm not a mathematician, and maybe all what I said is mathematically wrong, I let to the true mathematicians here to correct any stupid thing I could have said.

 

[BvU]

I want to understand this Set and variable x concept from scratch...

One way to do that is to accept it for what you think it is and move on. Experience will help you form a more detailed picture. At the moment you let yourself be blocked by something that will become more clear if you work with it in practice.

 

[PeroK]

I want to understand the full concept of the variable x,
what does it mean when we say x∈R ? I want understand the concept in detail.
if x is an element of the set R, then how can every element be x it is just like saying- {x,x,x,x,x,x,x}. That's why I am asking what does "for each x in ..." or "for every x in..." mean.
I want to understand this Set and variable x concept from scratch...
Help me...

I think if you delve into the foundations of mathematics these concepts become less clear, not any clearer. The basic starting point is what has been said in this thread.

Let me ask you a question. Let's suppose we are looking at the laws of cricket, and specifically Law 32 (Bowled):

A batsman is out if his wicket is put down by a ball delivered by the bowler.

What does it mean by "a batsman"? Is that a specific batsman? Is that all batsmen at the same time?

You tell us what you don't understand about that law of cricket.

 

[Mark44]

We don't. Based on
$x\in \mathbb R$ alone, we don't know anything else about $x$. It may not even exist.

x should represent a single real number right ?

Not necessarily:
$x\in \mathbb R\ \ \& \ \ x^2 = -1\qquad$ no solution.
$x\in \mathbb R \ \ \& \ \ x^2 = \ \ 1\qquad$ two solutions.
$x\in \mathbb R \ \ \& \ \ x>1 \ \ \&\ \ x < 1.0000000000000001\qquad$ inifnite number of solutions.

I don't agree with the above. If all that is stated is $x\in \mathbb R$, it's reasonable to assume that x exists and is some unspecified real number. However, if you add further restrictions or qualifications, as in the above, then it's possible that there is either no value of x that meets the new constraints, or that multiple values do so.

 

[Jarvis323]

$x$ is a variable.

It's sort of like a seat in a stadium. It doesn't represent a person per se. It represents a place that people can sit, sort of.

Saying $x \in \mathbb{R}$ is like saying only real numbers can sit in that seat/placeholder.

But when people do math, they aren't always completely precise in their language. It's just easier to act like $x$ is actually some number, rather than saying something like let a variable $x$ be substitutable by only real numbers, then some expression is true for substitutable numbers for $x$, or something like that.

Usually when people say let $x$ be <something>, they mean suppose there were an arbitrary number that <something> and let's call it $x$. This means that <something> is the only restriction on what can be substituted for $x$, so whatever you prove, is valid for every number that meets those restrictions.

 

[SHASHWAT PRATAP SING]

Let me tell you what I understand-
Often we want to to prove something about all elements of a set S. That is, we want to prove that every element in S satisfies some property, call it P. For example, say S={1,2,...}. Now say P(x) stands for the statement "x is greater than 0". Clearly P(1) is true, and so is P(2) is true, and indeed it should be obvious that no matter what number we choose from S,P( 'that number' ) will be true.

We want a way to express the truth of the statement in the above example. We know that no matter what object I choose from S, the statement P will be true for that object. We do this by reasoning about an arbitrary(any) object from that set. We say x∈S, and specify nothing more. Now the key bit here is that x is not simultaneously "all objects in S". x represents a specific(single) number from the set S, we just don't say which one.

Going back to our example, say I tell you x∈S. Now x represents some specific number in S, but we don't know which. Suppose x actually represented the number 20. Well of course P(20) is true, because 20>0. Similarly, if x actually represented the number 1727361, then P(1727361)would be true too. The pattern here is clear: no matter what number x actually represents, P(x) is true. It is in this sense that x can represent any element of S. So, we use 'any' in the sense that x is not specified,But we should keep in mind that x represents a specific (yet unspecified) number.
Although x is not specified that which number it represents yet x represents a specific(single) number which could be any number.
So, here I understood the meaning of 'any'. Please tell me am I right ?

I am confused with the use of "for all x in...".
I don't understand what do we mean when we say 'for all x in..." or "for each x in ..." I start to visulaise this as -> {x,x,x,x,x,x} something like this,
this is what I want to understand. please help me...

 

[Jarvis323]

That sounds basically right. But if you want to get even more precise, you need to understand what is meant by 'represents'. Technically x is not the element in the set, it is a placeholder that a member of the set can replace. Once you've substituted x for the element, then you have an expression that can be evaluated.

So I would say, x is neither a specific element, nor all elements in the set, x is a slot that elements of the set can be put in, one at a time.

 

[Jarvis323]

I am confused with the use of "for all x in...".
I don't understand what do we mean when we say 'for all x in..." or "for each x in ..." I start to visulaise this as -> {x,x,x,x,x,x} something like this,
this is what I want to understand. please help me...

If means that for every substitution of x by an element of this set. Each substitution produces a single statement. By using the variable we can say things about all of them at the same time. E.g. for all of the statements resulting from substituting x for an element of S , <something>

 

[Mark44]

I am confused with the use of "for all x in...".
I don't understand what do we mean when we say 'for all x in..." or "for each x in ..." I start to visulaise this as -> {x,x,x,x,x,x} something like this,

This part, {x, x, x, x, x, x}, doesn't make any sense. The elements in a set aren't repeated.
If S = {1, 2, 3, 4, ...} (i.e., S is the set of positive integers), the expression "for all x in S" means that x can be replaced by anyone of the numbers in S.

Equivalent notation is $\forall x \in S$.

 

[Mark44]

I want to understand the full concept of the variable x,
what does it mean when we say x∈R ? I want understand the concept in detail.

You are overthinking this very simple concept. A variable is a placeholder that can be replaced by some value. It works sort of like pronouns do in English. If I say "James is here today, but he won't be here tomorrow," the pronoun he can be understood to represent James in this sentence. However, in a different sentence, the word he could represent some other (male) person or animal.

Another example is the word "today." What the word "today" means depends on whether it is Monday right now or Tuesday or whatever.

 

[Stephen Tashi]

When we say that x∈R does here x represents all real numbers at the same time or x represents only a single real number at one time.
How can a single variable(x) represent all the real numbers at the same time ?

There is a principle in logic called "universal generalization". In writing proofs that employ this principle, writers introduce a "variable" such as "x" that is treated like a specific thing (for example, a specific number). Something is proved about "x", reasoning as if "x" represents a specific thing. No special properties are assumed for "x" that are not true of other things of the same type. The proof shows that the conclusion is true for all things of the same type as "x".

For example, in geometry, if some fact is to be proven about triangles, a proof may begin by saying "Let ABC be a triangle".

In terms of symbolic logic, to make a sentence be definitely true or false, it is necessary to put all "variables" within the scope of a quantifier. For example, "x is larger than 3" is a "propositional function" , not a "proposition" since it is only true or false once something specific is substituted for "x". To turn a propositional function into a (True or False) proposition, we put the variables of the propositional function in the "scope" of quantifiers. For example, we can say "There exists an x such that x is greater than 3" or "For each number x, x is greater than 3".

In writing about mathematics it is common to omit the the quantifier $\forall$ (for each...) and leave it as "understood". For example, someone who writes "$x + y = y + x$" may mean that "For each real number $x$ and each real number $y$, $x + y = y + x$.

The use of "universal generalization" is an exception to this pattern because we begin by introducing a variable "x" without quantifying it by "for each" or "there exists an". Some might argue that a proof that begins with "Let $x \in \mathbb{R}$" could be re-written as a proof that begins with "For each $x \in \mathbb{R}$...". However, some textbooks on logic treat "universal generalization" as special case where we are allowed to introduce a variable without putting it in the scope of a quantifier.

A technical statement of the principle: https://proofwiki.org/wiki/Universal_Generalisation

 

[nuuskur]

The element is always fixed. Your argument might not depend on the choice of $x$, so whatever you proved holds for every $x\in\mathbb R$.

why we start to treat it as if x represents all the real numbers at once as in inequality suppose we have x>-2 this means that x can be any real number greater than -2 but then why we say that all the real numbers greater than -2 are the solutions of the inequality

Given an inequality $f(x)>c$, after some work you determined that
$$\forall x\in\mathbb R,\quad x>-2 \Rightarrow f(x)>c.$$