up:: Equivalence Class
Let be an Equivalence Relation.
Let two equivalence classes, and suppose that there is some point in common to both.
Then and . But, by definition, .
Thus, all elements of are equivalent to , and thus .
up:: Equivalence Class
Let ∼ be an Equivalence Relation.
Let [x],[y] two equivalence classes, and suppose that there is some point z∈X in common to both.
Then ∀x∈X,x∼z and ∀y∈Y,y∼z⟺z∼y. But, by definition, x∼z∧z∼y⟹x∼y.
Thus, all elements of x are equivalent to y, and thus [x]=[y].