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.
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.
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.
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.
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.