We welcome our new author Nurlan Abiev. He intends to cover topics in Mathematics.
Under a set we mean an undefined term thought intuitively as a collection of objects of common property.
Definition 1. Let and be sets.
i) is said to be a subset of if each element of belongs to . Usually, this fact we denote by . Likewise, for every set .
ii) and are equal if they consist of the same elements. We say that is a proper subset of , if and . Consequently, we accept the notation .
iii) Obviously, a set consisting of no elements we call the empty set . Moreover, for any nonempty set we accept the agreement .
Definition 2. The product set of sets and is defined as a set of all ordered pairs , where and .
Example 1. Let and . Then we have . Further, note that , but .
Definition 3. Any subset of we call a relation between and , in other words .
Furthermore, we say that is in the relation to if , and denote this fact by . Respectively, the negation means .
Definition 4. An equivalence relation on a set is a relation satisfying the following conditions for arbitrary :
ii) If then (symmetry);
iii) If and then (transitivity).
In cases when is an equivalence relation we will use the notation instead of .
Example 2. Assume that is the set of all people in the world. Consider some relations on .
i) Descendant relation. A relation if is a descendant of is transitive, but not reflexive nor symmetric.
ii) Blood relation. A relation if has an ancestor who is also an ancestor of is reflexive and symmetric, but not transitive (I am not in a blood relation to my wife, although our children are).
iii) Sibling relation. Assuming if and have the same parents we define an equivalence relation.
Recall that and are the sets of reals and integers respectively.
Example 3. Examples of equivalence (non equivalence) relations:
i) For define a relation if . This is an equivalence relation on .
ii) Let if . This relation is not symmetric.
iii) For define if is even. Consequently we obtain an equivalence relation on .
iv) A relation if is odd can not be an equivalence relation on because it is not reflexive and not transitive.
v) Let . Clearly, if is not an equivalence relation since it is not transitive.
vi) Any function induces an equivalence relation on setting if .
Definition 5. Assume that is an equivalence relation on a given set . We say that is the equivalence class determined by under the equivalence relation .
The set of all equivalence classes we call the quotient set of over , in symbols .
Theorem 1. Every equivalence relation on a given set provides a partition of this set to a disjoint union of equivalence classes, in other words, if is an equivalence relation on then
Thus any two equivalence classes coincide if they admit a nonempty intersection.
i) By reflexivity for every . So .
ii) Let . Then and imply , showing .
By symmetry we also have . Then analogously. Therefore, .
iii) Suppose that , and let be their common element. According to symmetry and transitivity it follows then and , which implies .
Theorem 1 is proved.
The converse of Theorem 1 is also true.
Theorem 2. Every partition of a set to a disjoint union of its subsets defines on an equivalence relation if and belong to the same subset of .