Equivalence Relation: Partition of {a,b,c} - Andy

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Thread starter andydoinmath
Start date Aug 21, 2013
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Equivalence Equivalence relations Relations

In summary, 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. A partition is a way of dividing or separating a set into disjoint subsets or parts. To determine if a relation is an equivalence relation, you must check if it satisfies the three properties of reflexivity, symmetry, and transitivity. An equivalence relation and a partition are closely related because the equivalence classes of the relation form the subsets of the partition. One example of an equivalence relation is the "same parity" relation on the set of integers, which would have a corresponding partition of even and odd numbers. Another example is the "same nationality" relation on a group of

Aug 21, 2013

andydoinmath

If {{a,b},{c}} is the partition of {a,b,c}. When finding the equivalence relation used to generate a partition, is it enough to say {a,b}x{a,b} U {c}x{c}?

Thanks
Andy

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Aug 21, 2013

Evgeny.Makarov

Gold Member

MHB

Yes.

Related to Equivalence Relation: Partition of {a,b,c} - Andy

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. In simpler terms, it is a way of grouping elements of a set together based on some common characteristic or property.

2. What is a partition?

A partition is a way of dividing or separating a set into disjoint subsets or parts. In other words, it is a collection of subsets that together cover the entire original set and have no elements in common. Each subset is called an equivalence class and the collection of all equivalence classes forms a partition.

3. How do you determine if a relation is an equivalence relation?

To determine if a relation is an equivalence relation, you must check if it satisfies the three properties of reflexivity, symmetry, and transitivity. Reflexivity means that each element is related to itself, symmetry means that if one element is related to another, then the other element is also related to the first, and transitivity means that if one element is related to a second and the second is related to a third, then the first is also related to the third.

4. How is an equivalence relation related to a partition?

An equivalence relation and a partition are closely related because the equivalence classes of the relation form the subsets of the partition. Each subset in the partition is made up of elements that are related to each other through the equivalence relation. In other words, the partition organizes the elements of the set into groups based on the equivalence relation.

5. Can you give an example of an equivalence relation and its corresponding partition?

One example of an equivalence relation is the "same parity" relation on the set of integers. This relation groups together all the even numbers and all the odd numbers, as they have the same remainder when divided by 2. The corresponding partition of this set would have two subsets: one for even numbers and one for odd numbers. Another example is the "same nationality" relation on a group of people, where the equivalence classes would be based on the country they are from.