Let R be an equivalence relation on a set A. Prove the following statement directly from the definitions of equivalence relation and equivalence class. For every a and b in A, if [a] = [b] then a Rb. Some of the sentences in the following scrambled list can be used to prove the statement. By definition of equivalence class, a E[b]. Let b E[a]. Then by definition of equivalence class, b Ra. Since R is symmetric, if b R a then a R b. | Thus, a Ra, and so, by definition of equivalence class, a E [a]. Since [a] = [b] and a E [a], then a E [b] by definition of set equality. | Hence, by definition of equivalence class, a Rb. By definition of equivalence relation, R is reflexive. Proof: Construct a proof by selecting appropriate sentences from the list and putting them in the correct order. 1. Suppose R is an equivalence relation on a set A, a and b are in A, and [a] = [b]. 2. —Select— 3. —-Select— 4. —Select— 5. —Select—

equivalencerelation then for every a and b insetR , if a = b then aRb exists.relationon set A, a and b are in A and [a]=[b]reflexivesymmetric, transitive binary link. The relationship of equipollence between line segments in geometry is one example of an equivalence relation.classes. Two items in the given set are only comparable to one another if and when they belong to the same equivalence class.equivalencevisit: