Proof of Partition of a Set with Nested Partitions | Set Theory Proof 2

In summary: Aso x∈(Just one of B1,B2,B3 because disjoint by definition)case 1 x∈A You mean "x∈B1"so x∈ Just one of C11,C12 because disjoint by definitioncase 2 x∈BYou mean "x∈B2"so x∈ Just one of C21,C21 because disjoint by definitioncase 3 x∈CYou mean "x∈B3"so x∈ Just one of C31,C32 because disjoint by definitionwhich implies that x
  • #1
Jairo Rojas
17
0

Homework Statement


. Let A be a set and {B1, B2, B3} a partition of A. Assume {C11, C12} is a partition of B1, {C21, C22} is a partition of B2 and {C31, C32} is a partition of B3. Prove that {C11, C12, C21, C22, C31, C32} is a partition of A.

Homework Equations

The Attempt at a Solution


I know this problem looks a little bit intuitive but I just want to make sure I make sense
proof

so {C11, C12, C21, C22, C31, C32} is a partition of A because A is divided into 3 disjoint subsets which make a partition of A, B1,B2,B3. And then each subset of these 3 disjoint subsets is further divided into two disjoint subsets which make a partition of the parent subset. So all elements in the parent subsets will make a partition of A because every element in each parent subset are disjoint among the elements of every other parent subset.
 
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  • #2
Your idea is correct, but could be put in a bit more formal way, i.e., take an element in A and show that it must belong to one and only one of the sets Cij.
 
  • #3
Orodruin said:
Your idea is correct, but could be put in a bit more formal way, i.e., take an element in A and show that it must belong to one and only one of the sets Cij.
ok this is what I did
proof
x∈A
so x∈(Just one of B1,B2,B3 because disjoint by definition)
case 1 x∈A
so x∈ Just one of C11,C12 because disjoint by definition

case 2 x∈B
so x∈ Just one of C21,C22 because disjoint by definition

case 3 x∈C
so x∈ Just one of C31,C32 because disjoint by definition

which implies that x∈(Just one of C11,C12,C21,C22,C31,C32)
 
  • #4
Jairo Rojas said:
Just one of C11,C12 because disjoint by definition
I would also specify that x being in B1 rules out it being in any other C
 
  • #5
Jairo Rojas said:
ok this is what I did
proof
x∈A
so x∈(Just one of B1,B2,B3 because disjoint by definition)
case 1 x∈A
You mean "x∈B1"

so x∈ Just one of C11,C12 because disjoint by definition

case 2 x∈B
You mean "x∈B2"

so x∈ Just one of C21,C21 because disjoint by definition

case 3 x∈C
You mean "x∈B3"

so x∈ Just one of C31,C32 because disjoint by definition

which implies that x∈(Just one of C11,C12,C21,C22,C31,C32)
 
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  • #6
HallsofIvy said:
You mean "x∈B1"You mean "x∈B2"You mean "x∈B3"
yes, didn't notice that. Is the logic right?. How does this show that these elements make a partition of A?
 

FAQ: Proof of Partition of a Set with Nested Partitions | Set Theory Proof 2

1. What is the concept of nested partitions in set theory?

Nested partitions in set theory refer to a way of dividing a set into smaller subsets, where each subset contains elements that are also included in the larger set. This creates a hierarchical structure, with the largest set being divided into smaller subsets, and those subsets being further divided into even smaller subsets.

2. How does Proof of Partition of a Set with Nested Partitions apply to set theory?

This proof demonstrates the concept of nested partitions by showing that a set can be divided into smaller subsets, and those smaller subsets can be further divided into even smaller subsets, and so on. This helps to show the structure and organization of a set, and how it can be broken down into smaller, more manageable parts.

3. Why is Proof of Partition of a Set with Nested Partitions important in set theory?

This proof is important because it helps to illustrate the fundamental concept of a set being divided into smaller subsets, which is a crucial idea in set theory. It also helps to demonstrate the hierarchical structure of sets and how they can be organized into smaller and smaller subsets.

4. Can you give an example of nested partitions in set theory?

One example of nested partitions in set theory is the set of real numbers. This set can be divided into subsets such as positive and negative numbers, which can then be further divided into rational and irrational numbers. The rational numbers can then be divided into integers, fractions, and decimals, and so on.

5. How is Proof of Partition of a Set with Nested Partitions used in other areas of science?

The concept of nested partitions is not only used in set theory, but it is also applied in other areas of science such as biology, computer science, and physics. In biology, nested partitions are used to classify species into smaller groups based on their characteristics. In computer science, nested partitions are used in data structures to organize and store information. In physics, nested partitions are used to understand the hierarchical structure of matter and energy in the universe.

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