Are there sets that are not partitionable in certain ways?

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In summary, there are sets that are not partitionable in certain ways, such as the set \mathbb{R} which cannot be partitioned into a collection of lines of slope 2 in \mathbb{R}^2. However, it is possible to partition \mathbb{R} into singletons and \mathbb{R}^2 into lines of slope 2. The partition for \mathbb{R} is \cup_{x \in \mathbb{R}} U_x = \mathbb{R}, where U_x = \{x\}. The partition for \mathbb{R}^2 is \cup_{a \in \mathbb{R}} U_a
  • #1
quasar987
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Are there sets that are not partitionable in certain ways? For exemple, can I partition [itex]\mathbb{R}[/itex] into a collection of singletons?

Can I partition [itex]\mathbb{R}^2[/itex] into a collection of lines of slope 2?

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

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

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

Then take [tex](x_0, y_0) \in \mathbb{R}^2[/tex] and notice that [tex](x_0, y_0) \in U_{y_0 - 2x_0}[/tex]. Thus [tex]\cup_{a \in \mathbb{R}} U_a = \mathbb{R}^2[/tex]. A simple calculation will also reveal that [tex]U_a[/tex] and [tex] U_b[/tex] are either equal or disjoint (for any real a, b).
 
  • #3


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.
 

Related to Are there sets that are not partitionable in certain ways?

1. What do you mean by "partitionable" sets?

"Partitionable" sets refer to the ability to divide a set into smaller, non-overlapping subsets, also known as partitions. A set is considered to be partitionable if it can be divided into subsets that cover all the elements of the original set without any overlap.

2. Are there any sets that are not partitionable at all?

Yes, there are sets that are not partitionable. One example is the set of real numbers, which cannot be partitioned into smaller subsets without leaving some numbers out. This is because the real numbers form a continuous and infinite set, making it impossible to divide it into finite subsets without leaving gaps.

3. Can a set be partitionable in one way but not in another?

Yes, a set can be partitionable in some ways but not in others. For example, the set of natural numbers can be partitioned into subsets of even and odd numbers, but it cannot be partitioned into subsets of multiples of 3 and multiples of 5 without overlap.

4. How do you determine if a set is partitionable?

To determine if a set is partitionable, one can try to divide the set into smaller subsets without any overlap. If this is possible, then the set is partitionable. However, if there is no way to divide the set without overlap, then the set is not partitionable.

5. Why is the concept of partitionable sets important in mathematics?

The concept of partitionable sets is important in mathematics because it helps us understand the structure and properties of sets. It also has applications in various fields such as probability, topology, and computer science. Partitionable sets also play a crucial role in the study of infinite sets and their cardinality.

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