Partition induced by a relation

[chubbs145]

Let A be a nonempty set and fix the set B where B is a subset of A. Define the relation R on the power set of A as follows: for all subsets X, Y of A, X R Y if and only if B intersect X = B intersect Y.

Suppose we make some sets...

A={1,2,3} and B={1,2}

How do i find the partition of the power set of A? Do I need to know what X and Y are first?

 

[Swlabr]

Let A be a nonempty set and fix the set B where B is a subset of A. Define the relation R on the power set of A as follows: for all subsets X, Y of A, X R Y if and only if B intersect X = B intersect Y.

Suppose we make some sets...

A={1,2,3} and B={1,2}

How do i find the partition of the power set of A? Do I need to know what X and Y are first?

X and Y are just elements of the powerset of A. You need to find which elements of the powerset are "equal" using this relation.

So, for example, {1}R{1, 3}.

 

[undefined]

Let A be a nonempty set and fix the set B where B is a subset of A. Define the relation R on the power set of A as follows: for all subsets X, Y of A, X R Y if and only if B intersect X = B intersect Y.

Suppose we make some sets...

A={1,2,3} and B={1,2}

How do i find the partition of the power set of A? Do I need to know what X and Y are first?

A relation partitions a set if and only if it is an equivalence relation. The partitions are known as equivalence classes. If you've worked with congruence classes (mod n), then you will have a good idea of this concept from experience.

Wikipedia states it thus: "For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from any partition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the same part in P. Thus the notions of equivalence relation and partition are essentially equivalent." (source)

So we know beforehand from the problem statement that we are dealing with an equivalence relation, but we could verify it if we wanted.

But anyway, I'd like to consider a larger example to better illustrate how this applies.

A = {1,2,3,4,5,6}
B = {1,2,3}

Now notice the powerset of B has 2^3 elements, and each of these sets defines an equivalence class. So for example, {1,2} is in the powerset of B. Consider

C = {1,2,5}
D = {1,2,5,6}
E = {1,2,4}
F = {1,2}

These are all equivalent under the relation R.

Does that make sense?

 

[tonio]

Let A be a nonempty set and fix the set B where B is a subset of A. Define the relation R on the power set of A as follows: for all subsets X, Y of A, X R Y if and only if B intersect X = B intersect Y.

Suppose we make some sets...

A={1,2,3} and B={1,2}

How do i find the partition of the power set of A? Do I need to know what X and Y are first?

"To know X,Y first"? you need to check which subsets of A are related according to the given relation...for example, \(\displaystyle \{2\}R\{2,3\}\) , since

\(\displaystyle \{2\}\cap \{1,2\}=\{2,3\}\cap\{1,2\}=\{2\}\) . From here, both \(\displaystyle \{2\}\,,\,\{2,3\}\) will be in the same partition set (i.e., equivalence class) of the power set of A, since they're r-related.

Well, now you find all the partitions sets (=equiv. classes) one by one. After all, there're only 8 subsets here...(Wink)

Tonio

 

[chubbs145]

Yeah a little, I guess im not really seeing the difference between equivalence class and partition. Are you able to have equivalence classes in a partition of something like the powerset of A by the definition of R?

 

[tonio]

A relation partitions a set if and only if it is an equivalence relation. The partitions are known as equivalence classes. If you've worked with congruence classes (mod n), then you will have a good idea of this concept from experience.

Wikipedia states it thus: "For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from any partition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the same part in P. Thus the notions of equivalence relation and partition are essentially equivalent." (source)

So we know beforehand from the problem statement that we are dealing with an equivalence relation, but we could verify it if we wanted.

But anyway, I'd like to consider a larger example to better illustrate how this applies.

A = {1,2,3,4,5,6}
B = {1,2,3}

Now notice the powerset of B has 2^3 elements, and each of these sets defines an equivalence class. So for example, {1,2} is in the powerset of B. Consider

C = {1,2,5}
D = {1,2,5,6}
E = {1,2,4}
F = {1,2}

These are all equivalent under the relation R.

Does that make sense?

Perhaps it makes sense but it is, I'm afraid, wrong (or I am: not that many choices here (Giggle) ): for example, where does the set \(\displaystyle \{3\}\) belong to? Nowhere, according to you!

We have \(\displaystyle \{3\}\cap B=\{3\}\) , whereas none of the sets C,D,E,F intersects B in 3 only...

Tonio

 

[undefined]

Perhaps it makes sense but it is, I'm afraid, wrong (or I am: not that many choices here (Giggle) ): for example, where does the set \(\displaystyle \{3\}\) belong to? Nowhere, according to you!

We have \(\displaystyle \{3\}\cap B=\{3\}\) , whereas none of the sets C,D,E,F intersects B in 3 only...

Tonio

I'm not sure what you mean

{3} is in the powerset of B.

So it is in the partition described completely by

{3} 000
{3,6} 001
{3,5} 010
{3,5,6} 011
{3,4} 100
{3,4,6} 101
{3,4,5} 110
{3,4,5,6} 111

where I wrote binary numbers alongside to show how I was choosing elements from A that are not in B. So the above 8 sets all have the same intersection with B, so are equivalent under R, and there are no other sets in this partition.

Am I wrong?

 

[lozts]

see below

 

[lozts]

Let A be a nonempty set and fix the set B where B⊆A. Define the relation R on ℘(A), the power set of A, as follows: for all subsets X,Y of A, X R Y iff B∩X = B∩Y.

Suppose we make some sets...

A={1,2,3}
B={1,2}

How do i find the partition of the power set of A? Do I need to know what X and Y are first?

well we can find that:

℘(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

therefore the partition of the power set of A would be:

[{},{3}],
[{1},{1,3}],
[{2},{2,3}],
[{1,2},{1,2,3}]

reasoning:
[{},{3}]

if X={}, Y={} and B={1,2}
B∩X = B∩Y = {} = {} which is true

if X={}, Y={3} and B={1,2}
B∩X = B∩Y = {} = {} which is true

if X={3}, Y={3} and B={1,2}
B∩X = B∩Y = {} = {} which is true

if X={3}, Y={} and B={1,2}
B∩X = B∩Y = {} = {} which is true

~~~~~~~~~~~~~~~~~~~~~~~~~~
[{1},{1,3}]

if X={1}, Y={1} and B={1,2}
B∩X = B∩Y = {1} = {1} which is true

if X={1}, Y={1,3} and B={1,2}
B∩X = B∩Y = {1} = {1} which is true

if X={1,3}, Y={1} and B={1,2}
B∩X = B∩Y = {1} = {1} which is true

if X={1,3}, Y={1,3} and B={1,2}
B∩X = B∩Y = {1} = {1} which is true

~~~~~~~~~~~~~~~~~~~~~~~~~~

similarly for [{2},{2,3}] and [{1,2},{1,2,3}]

 

[Plato]

Let A be a nonempty set and fix the set B where B is a subset of A. Define the relation R on the power set of A as follows: for all subsets X, Y of A, X R Y if and only if B intersect X = B intersect Y.

This is a well-known problem. Let’s say that \(\displaystyle B\not= \emptyset\)
Then as was pointed out the equivalence classes are determined by the elements in the power set \(\displaystyle \mathcal{P}(B)\).
\(\displaystyle \forall X \in\mathcal{P}(B)\) however one denotes the equivalence class \(\displaystyle [X]= \left\{ {X \cup Y:Y \in \mathcal{P}(A\backslash B)} \right\}\)