What are the distinct equivalence classes of R?

There are five distinct equivalence classes, modulo 5: [0], [1], [2], [3], and [4]. {x ∈ Z | x = 5k, for some integers k}. Definition 5. Suppose R is an equivalence relation on a set A and S is an equivalence class of R.

How do you describe an equivalence class?

An equivalence class is the name that we give to the subset of S which includes all elements that are equivalent to each other. ‘The equivalence class of a consists of the set of all x, such that x = a’. In other words, any items in the set that are equal belong to the defined equivalence class.

How do you show equivalence relation in R?

To prove R is an equivalence relation, we must prove R is reflexive, symmetric, and transitive. So let a, b, c ∈ R. Then a − a = 0=0 · 2π where 0 ∈ Z. Thus (a, a) ∈ R and R is reflexive.

How do you find the number of distinct equivalence classes?

Finding how many distinct equivalence classes there are.

1. Hint: take any integer, say 0. Find all the elements equivalent to 0. They form an equivalence class. Take any integer not in this equivalence class and repeat.
2. Write x=3a+r,y=3b+s where r,s are the remainders when dividing x,y by 3. endgroup. – Jens Schwaiger.

What is equivalence relation explain with example?

An equivalence relation is a relationship on a set, generally denoted by “∼”, that is reflexive, symmetric, and transitive for everything in the set. Example: The relation “is equal to”, denoted “=”, is an equivalence relation on the set of real numbers since for any x, y, z ∈ R: 1. (Reflexivity) x = x, 2.

Which of the following properties are satisfied by equivalence relationship?

Equivalence relations are relations that have the following properties: They are reflexive: A is related to A. They are symmetric: if A is related to B, then B is related to A. They are transitive: if A is related to B and B is related to C then A is related to C.

How would you describe an equivalence relation?

Definition 1. An equivalence relation is a relationship on a set, generally denoted by “∼”, that is reflexive, symmetric, and transitive for everything in the set. Example: The relation “is equal to”, denoted “=”, is an equivalence relation on the set of real numbers since for any x, y, z ∈ R: 1.

How do you show that two equivalence classes are equal?

For each a,b∈A, a∼b if and only if [a]=[b]. Two elements of A are equivalent if and only if their equivalence classes are equal. Any two equivalence classes are either equal or they are disjoint. This means that if two equivalence classes are not disjoint then they must be equal.