Let $I:=[0, 1]$ be the unit interval. Can we construct a partition $\mathbf{P}=\{A_\alpha\}_{\alpha}$ with $\mid A_\alpha\mid=2$, that is, for each $\alpha$, $A_\alpha $ has exactly two elements?
A special partition for unit interval
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elementary-set-theory
2 Answers
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For each $r\in [0, \frac{1}{2})$ set $A_r=\{r, r+\frac{1}{2}\}$ is partition for the interval $[0, 1)$, and since $[0, 1)\approx [0, 1]$. Thus, $[0, 1]$ has a partition with mentioned property.
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Yes, you can partition $[0,1]$ into sets of size two. There isn't really a good way to write it though, and I would probably just refer to it as a partition of sets of size two.