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).