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When you write $x\in R$, this reads $x$ is an element of $R$. This specifies a domain for $x$ which is simply a variable.

When you define a set $X = \{x\}$ , the set of all $x$, or $X = \{x\in R\}$ it seems as though to define the set you are specifying a domain for $x$, the values $x$ can take is the set. My question is how would you refer to $x$? I know you can refer to it as the general element but I can't really understand this.

Henrik supports the community
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Carlos Bacca
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1 Answers1

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$X = \{x\}$ says $X$ is a set and it has one element in it.

If you want to define a set $X$ and then say for all $x\in X$ that is fine. You don't need anything else other than "for set $X$" or similar. Heck, you often don't even need to say that much.

And if you need to be more specific, you may want to say that $X$ is a subset of the natural numbers. i.e. $X\subset \mathbb N$

And maybe you want to define a set like the set of point on the unit circle. $X=\{(x,y)|x^2 + y^2 = 1 \}$

Doug M
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  • Thanks, if your last line defines the set of points on the unit circle then why does X = {x} not define the set of all x because your not placing any restriction on x? – Carlos Bacca Aug 14 '18 at 20:51
  • Then how would you define the set which contained the element x? I do see your point. Nonetheless, if you want $X$ to be most generically a set, then just say so. If you want to put conditions on the elements then define the conditions. – Doug M Aug 14 '18 at 21:41
  • You have to impose conditions, because there is no "set of everything". That would actually result in logical errors if there was such a thing. – red_trumpet Aug 14 '18 at 21:41
  • You might also see something line ${a}$ is a sequence converging to $a.$ And sequences use many of the concepts and vocabulary as sets do. – Doug M Aug 14 '18 at 21:43
  • @red_trumpet you might say $f:X\to Y$ is a function. $X,Y$ are sets, but we might never get around to their particulars. But we must assume that do meet the ZF requirements of a set (i.e. not the set of all sets, or similar) – Doug M Aug 14 '18 at 21:48