Is this Partition affirmation correct? [closed]
This is the sentence.
If {$A_i$}$_i$$_∈$$_\mathbb{Z}$ is a partition of $\mathbb{R}$ then {$\sqrt{2}$} $⊆ A_i$ for one and only one value of $i$.
I guess it's true, because each partition differs from one another. Is this Partition affirmation correct?
Solution 1:
It's true, since in each partition, (1) the sets $A_i$ are nonempty, (2) their union is the ambient set $\Bbb R$, and (3) two sets are either equal or disjoint.
Condition 2) ensures that $\sqrt 2\in\Bbb R$ lies in one of the sets $A_i$, and condition 3) shows that the set $A_i$ is uniquely determined.