As usual, let’s start with the relevant definitions.
Definition: “A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.”
Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.”
Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R, for all a,b ∈ A.”
Definition: “A relation R on a set A is called transitive if whenever (a, b)∈R and (b, c)∈R, then (a, c) ∈ R, for all a,b,c ∈ A.”
The definitions above are from the textbook “Discrete Mathematics and its applications” By Rosen.
In this case, we can find a function f, such that (x,y) ∈ R if and only if f(x) = f(y)
We can define the function as follows:
- f(x)=[x]R
Notice that this function has A as its domain. The range is the set of all equivalence classes over A defined by R.
So, the function will map each element x∈A to the equivalence class of such element defined by R: [x]R.
So, ∀x,y∈A, (x,y) ∈ R if and only if f(x) = f(y).
⧠
Notice that a relation R={(a,b)| [a]=[b]} is an equivalence relation.
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