Equivalence Relations and Quotient Sets - Verifying a Claim

In summary, the quotient set of a set S relative to an equivalence relation on S is not a subset of S. It is the set of all equivalence classes defined by the relation and can be identified with a subset of S, but this identification can vary depending on the choices of "representative".
  • #1
matheater
7
0
I have a question...
"Is the quotient set of a set S relative to a equivalence relation on S a subset of S?"
I suppose "no",since the each member of the quotient set is a subset of S and consequently it is a subset of the power set of S,but I have e book saying that "yes",I am a bit confused,can anyone ensure me?
 
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  • #2
No, the quotient set of set S, relative to some equivalence relation on S, is not a subset of S. It is the set of all equivalence classes defined by the relation and so, as you say, a subset of the power set of S. We could, by choosing one "representative" of each equivalence class, "identify" the quotient set with a subset of S but that can be done in many different ways depending on the choices of "representative".

For example, if S= N, the natural numbers, and the relation is xRy if and only if x-y is a multiple of 3, then the quotient set is the set containing: the multiples of 3, the set of numbers of the form 3n+1, and the set of numbers of the form 3n+2. We can, and often do identify those with {0, 1, 2}. But we could as easily identify them with {3, 4, 5}, etc.
 
  • #3
Thank u very much,I am complete agree with u.
 

Related to Equivalence Relations and Quotient Sets - Verifying a Claim

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that defines a relationship between elements of a set. It is a binary relation that satisfies three properties: reflexivity, symmetry, and transitivity. This means that for any elements a, b, and c in the set, if a is related to b and b is related to c, then a is also related to c. Equivalence relations are used to group elements into equivalence classes based on their shared properties or characteristics.

2. How do you verify a claim about an equivalence relation?

To verify a claim about an equivalence relation, you need to check if it satisfies the three properties: reflexivity, symmetry, and transitivity. This can be done by examining the definition of the relation and using examples to see if the properties hold. If all three properties are satisfied, then the claim is true and the relation is an equivalence relation.

3. What is a quotient set?

A quotient set, also known as a factor set, is a set that contains the equivalence classes of an equivalence relation. It is created by dividing a set into subsets based on the relation and then grouping together all the elements that are related to each other. The quotient set allows us to simplify and understand the structure of a larger set by categorizing its elements into smaller, more manageable subsets.

4. How do you prove that two quotient sets are equal?

To prove that two quotient sets are equal, you need to show that their elements are the same. This means that every element in one quotient set is also in the other quotient set, and vice versa. This can be done by showing that the equivalence classes in both sets are the same, which can be done by checking if the elements in each class are related to each other and if there are any missing elements.

5. What are some real-world applications of equivalence relations and quotient sets?

Equivalence relations and quotient sets have many applications in various fields of mathematics and science. In computer science, they are used in data structures and algorithms to efficiently group and sort data. In graph theory, they are used to model connections and relationships between objects. In physics, they are used to study symmetries and conservation laws. In social sciences, they are used to classify and analyze data in fields such as economics, psychology, and sociology. Equivalence relations and quotient sets also have practical applications in engineering, statistics, and economics.

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