What Determines the Number of Equivalence Classes in a Set?

• cragar
In summary, when considering the word "mississippi", there are 11 elements in the multiset and 4 equivalence classes based on the 4 different letters. However, when dealing with the set A={1,2,3}, the number of equivalence classes depends on the defined equivalence relation. It is important to clarify the equivalence relation when discussing equivalence classes.
cragar

Homework Statement

Lets say I have the word mississippi .
Would I then say that I have 11 elements in my multiset .
And would I say that I have 4 equivalence classes because I only have 4 different letters.

If I had the set A={1,2,3,} Would I say this has 3 different equivalence classes.

i think you need to define exactly what your set is and what the equivalence class is defined by

an example might be the set of numbers
{2,3,4}

we could partition into 2 equivalence classes could be whether or not the number is divisible by 2

the letter case a little confusing as there are repeated elements in the set

Last edited:
cragar said:

Homework Statement

Lets say I have the word mississippi .
Would I then say that I have 11 elements in my multiset .
Yup, if each element of the multiset is a single letter.
And would I say that I have 4 equivalence classes because I only have 4 different letters.

If I had the set A={1,2,3,} Would I say this has 3 different equivalence classes.
Not necessarily. The equivalence classes depend on exactly what equivalence relation you have.

It makes no sense at all to talk about "equivalence classes" without stating the equivalence relation. You have two questions here. What are your equivalence relations?

1. What are equivalence classes?

Equivalence classes are a mathematical concept used to group together elements that have the same characteristics or properties. In other words, they are sets of objects that are considered equivalent to each other.

2. How are equivalence classes determined?

Equivalence classes are determined based on a specific equivalence relation. This relation defines what characteristics or properties are used to group elements together. For example, if we have an equivalence relation based on the color of objects, we can group all red objects together in one equivalence class, all blue objects in another, and so on.

3. Can elements belong to multiple equivalence classes?

No, elements can only belong to one equivalence class at a time. This is because equivalence classes are mutually exclusive and exhaustive, meaning that all elements are placed in one and only one equivalence class based on the defined equivalence relation.

4. How are equivalence classes represented?

Equivalence classes can be represented in various ways, depending on the context. In mathematics, they are often denoted by brackets or curly braces enclosing the elements in the class. In computer science, they can be represented using data structures such as arrays or linked lists. In other fields, they may be represented using different symbols or notations.

5. What is the significance of equivalence classes?

Equivalence classes are important in various fields, including mathematics, computer science, and statistics. They help us to organize and classify objects based on their properties, making it easier to analyze and understand complex systems. In addition, they are used in various algorithms and data structures to efficiently process and manipulate data.

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