Are there sets that are not partitionable in certain ways?

[quasar987]

Are there sets that are not partitionable in certain ways? For exemple, can I partition $\mathbb{R}$ into a collection of singletons?

Can I partition $\mathbb{R}^2$ into a collection of lines of slope 2?

If so, how would you write each of those partitions?

Thx.

 

[Muzza]

For the first one: for every $$x \in \mathbb{R}$$ let $$U_x = \{x\}$$. Then $$\cup_{x \in \mathbb{R}} U_x = \mathbb{R}$$ and obviously all $$U_x$$ are disjoint from one another, and are singletons.

For the second: for every $$a \in \mathbb{R}$$ let $$U_a = \{(x, y) | y = 2x + a\}$$ (I wish I knew LaTeX better).

Then take $$(x_0, y_0) \in \mathbb{R}^2$$ and notice that $$(x_0, y_0) \in U_{y_0 - 2x_0}$$. Thus $$\cup_{a \in \mathbb{R}} U_a = \mathbb{R}^2$$. A simple calculation will also reveal that $$U_a$$ and $$U_b$$ are either equal or disjoint (for any real a, b).

 

[tacman]

Yes, there are sets that are not partitionable in certain ways. A set is partitionable if it can be divided into non-empty subsets such that each element in the original set belongs to exactly one of the subsets. However, not all sets can be partitioned in certain ways, as shown in the examples given.

For the first example, \mathbb{R} can be partitioned into singletons, which are sets containing only one element. This can be written as \mathbb{R} = \{x\} for all x \in \mathbb{R}. Each singleton subset would contain only one real number, and together they would cover all real numbers.

On the other hand, \mathbb{R}^2 cannot be partitioned into lines of slope 2. This is because a line of slope 2 would contain points with coordinates (x, 2x), where x is any real number. However, there are infinitely many points in \mathbb{R}^2 that do not have coordinates in the form (x, 2x), making it impossible to cover the entire set with lines of slope 2. Therefore, this set is not partitionable in this particular way.

In general, the ability to partition a set depends on the specific criteria or conditions set for the subsets. While some sets can be easily partitioned, others may not have a clear or feasible way of being divided into subsets.