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When we say that $x \in \mathbb R$ 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 $x \in \mathbb R$ 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 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 number.Please help me....

luxerhia
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Shashwat singh
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2 Answers2

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When we say $x\in \mathbb R$, we mean that $x$ is an arbitrary element in $\mathbb R$. That doesn't mean to "represent all the real numbers at the same time", but to present any numbers. Usually we will have a proposition including $x$, and it is true if and only if you choose any real values for $x$, the proposition always holds.

FFjet
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A sentence like

Let $x \in E$. Suppose $x$ satisfies $blahblahblah$. Then $P$ is true.

means

$\forall x,\quad (x \in E \mbox{ and } blahblahblah) \Rightarrow $P$.

The rest is irrelevant: "represent" is not a mathematical word, so it doesn't make sense to wonder about how a single letter could represent all real numbers.

I think that sometimes, the teachers express themselves in a strange way, using the word "variable" which has no formal sense either, etc. and it confuses students; in any case, I think it is a healthy reaction to get confused by these things!

Plop
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