Fundamental Theorem of Equivalence Relations: An equivalence relation is a binary relation (R) on a set X that is reflexive, symmetric, and transitive....
Fundamental Theorem of Equivalence Relations:
An equivalence relation is a binary relation (R) on a set X that is reflexive, symmetric, and transitive.
Reflexivity: For all elements a, b in X, if a R b, then b R a.
Symmetry: If a R b and b R c, then a R c.
Transitivity: If a R b and b R c, then a R c.
In other words, an equivalence relation is a relationship that "mirrors" the structure of the set. It tells us that if we take any two elements a and b and think of all the elements that are related to both of them, they will be related to each other as well.
Furthermore, an equivalence relation is reflexive if and only if the relation is symmetric. If a R b and b R c, then a R c.
Examples:
The relation "is equal to" on the set of real numbers is an equivalence relation.
The relation "is divisible by" on the set of integers is an equivalence relation.
The equivalence relation "is congruent to" on the set of 2D shapes is an equivalence relation.
The fundamental theorem of equivalence relations ensures that equivalence relations are the building blocks of equivalence partitions, which are subsets of the set that are "equivalent" to each other. Equivalence partitions are used in various mathematical fields, including topology, graph theory, and dynamical systems