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'A or B' means one or the other or both.

So I think $\{x | x = u$ or $ x = v\}$ can be equal to $\{u\}$ or $\{v\}$ or $\{u, v\}$.
Similary, I think $\{x | x\in a $ or $ x\in b\}$ can be equal to any subset of $a \cup b$ except empty set.

But actually $\{x | x = u$ or $ x = v\} = \{u, v\}$ and $\{x | x\in a $ or $ x\in b\} = a \cup b$. What am I missing?

op ol
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    ${ x \in \mathbb R \mid x = 1 \text{ or } x = 2 }$ consists of all real numbers $x$ that satisfy either $x = 1$ or $x = 2$. – littleO Feb 13 '21 at 07:36
  • @littleO Ok, I got what you said. But I think my interpretation also makes sense. Isn't it? Is their a definition about how to use the term 'or'? – op ol Feb 13 '21 at 07:41
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    ${x|x=u\text{ or }x=v}$ is the set of ALL $x$ such that $x=u$ or $x=v$. Note that both $u$ and $v$ satisfy this property. So they are both in the set. Similarly, ${x|x\in A\text{ or }x\in B}$ is the set of ALL $x$ belonging to either $A$ or $B$ (possibly both). Every element of $A$ and $B$ satisfies this, so you get the whole union $A\cup B$. – morrowmh Feb 13 '21 at 07:45
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    Let $S = { x \in \mathbb R \mid x = 1 \text{ or } x = 2 }$. Does $1$ belong to $S$? Yes, it satisfies the condition necessary to belong to $S$. Does $2$ belong to $S$? Yes, it also satisfies the condition necessary to belong to $S$. – littleO Feb 13 '21 at 08:00
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    "But I think my interpretation also makes sense. " It makes sense, yes, but unfortunately it is incorrect. You are pre-supposing that there is a number $x$ and that it currently exists and we are being told something about it. Set notation means the $x$ is a label for all potential values and all possible values are in the set. ... consider ${x|x\in \mathbb R}$. By your interpretation $5 \in \mathbb R$ so $x$ could be $5$. So the set could be ${5}$. That's... consistent but pointless. Instead the set is all possible things that $x$ could be. i.e ${x|x\in \mathbb R}=\mathbb R$. – fleablood Feb 13 '21 at 08:57

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$x$ is variable and is a hypothetical. It is a place value of everything that could be. It is not an actual thing and doesn't actually equal anything.

$\{x|x= u$ or $x=v\} = \{$ all possible numbers so long as the number we are considering is either $u$ or $v\}$.

As $u$ is a possible value $u$ must bein the set. And as $v$ is a possible value it is in the set. Any other value is neither $u,v$ so no other value is in the set.

So the set $\{x|x=u$ or $x=v\}$ is precisely $\{u,v\}$.

fleablood
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It sounds like what you're missing is that your example uses the set builder notation $\{x|P(x)\}$, which is defined to mean "the set of all objects $x$ for which the sentence $P(x)$ is true". The "or" in your example is part of the $P(x)$ formula - it's incorrect to interpret the expression in a different way (such as the way you might interpret an English sentence with "or" in it).

Karl
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