My lecture notes say that the class of this has one element, i.e.,$\left[ x\right] =\left\{ x\right\}$. Why? Can you explain?
What can we say about the class of $x\equiv y\Leftrightarrow x=y$ for all $x,y\in\mathbb{Z}$?
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elementary-set-theory
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1Suppose it had more than one element. Then there are at least two different elements., call them $x\in [x]$ and $y\in [x]$. That implies since they both belong to the same class that they are "equivalent", i.e. $x\equiv y$. But by the definition given for equivalent in this context, that implies that $x=y$, and we see that they are in fact not different elements at all but the same element. – 2017-01-09
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1Did you mean to ask: For all $x, y \in \mathbb Z$, $x\equiv y \iff x= y$.? – 2017-01-09
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0@amWhy Yes, I did. – 2017-01-09
1 Answers
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Using the definition of equivalence class: $$[x]\underbrace{=}_{\text{def}}\{y\in\mathbb{Z}:y\equiv x\}=\{y\in\mathbb{Z}:y=x\}=\{x\}.$$