Define a relation ~ on P(X), where X = ForY, Z e P(X), we say Y ~ Z if and only if Y has the same number of ele-ments as Z.{1,2, 3, 4, 5, 6}, as follows:List and describe all unique equivalence classes.

Given a set X={1,2,3,4,5,6}. Define a relation ~ on P(X) as : For Y,Z∈ P(X), we say Y~Z if and only…

Answered: on P(X), where X = Y, Z E P(X), we say…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 23E: 23. Let be the equivalence relation on defined by if and only if there exists an element in ...

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Define a relation ~ on P(X), where X = For
Y, Z e P(X), we say Y ~ Z if and only if Y has the same number of ele-
ments as Z.
{1,2, 3, 4, 5, 6}, as follows:
List and describe all unique equivalence classes.

Expert Solution

Step 1

Given a set $X = {1, 2, 3, 4, 5, 6}$ .

Define a relation $~$ on $P (X)$ as :

For $Y, Z \in P (X)$ , we say $Y ~ Z$ if and only if Y and Z has same number of elements.

Means the cardinality of set Y is equal to the cardinality of set Z,i.e. $|Y| = |Z|$ .

We know, $~$ on $P (X)$ is an equivalence relation.

So,for each set A in $P (X)$ , the equivalence classes of the set A denoted as [A] and defined as:

$[A] = {B \in P (X) : A ~ B}$

Step 2

$X = {1, 2, 3, 4, 5, 6}$

The power set of X is

$P (X) = {\emptyset, {1}, {2}, {3}, {4}, {5}, {6}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}, {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 2, 6}, {1, 3, 4}, {1, 3, 5}, {1, 3, 6}, {1, 4, 5}, {1, 4, 6}, {1, 5, 6}, {2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 4, 5}, {2, 4, 6}, {2, 5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 3, 6}, {1, 2, 4, 5}, {1, 2, 4, 6}, {1, 2, 5, 6}, {1, 3, 4, 5}, {1, 3, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {2, 3, 4, 5}, {2, 3, 4, 6}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 3, 5, 6}, {1, 2, 4, 5, 6}, {1, 3, 4, 5, 6}, {2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6}}$

Now, we will define equivalence classes of each set in $P (X)$ .

$[\emptyset] = {\emptyset}$ ,

$[{1}] = {{1}, {2}, {3}, {4}, {5}, {6}}$ ,

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