Pembuktian Kalkulus Bab Himpunan

Jika P, Q, R, adalah himpunan, buktikan bahwa :

(P U Q) ∩ (P’ ∩ R)’ = P U (Q’ U R)’

Misalkan :               S = {x │ x bilangan bulat positif, x ≤ 7}

                                P = {1, 2, 4, 6}

                                Q = {2, 4, 6}

                                R = {4, 5, 6, 7}

(P U Q) ∩ (P’ ∩ R)’  = P U (Q’ U R)’

P U Q                                = {1, 2, 4, 6} U {2, 4, 6}

                                           = {1, 2, 4, 6}

P’                                        = {3, 5, 7}

P’ ∩ R                                = {3, 5, 7} ∩ {4, 5, 6, 7}

                                            = {5, 7}

(P’ ∩ R)’                             = {1, 2, 3, 4, 6}

(P U Q) ∩ (P’ ∩ R)’           = {1, 2, 4, 6} ∩ {1, 2, 3, 4, 6}

                                             = {1, 2, 4, 6}

Q’                                         = {1, 3, 5, 7}

Q’ U R                                 = {1, 3, 5, 7} U {4, 5, 6, 7}

                                             = {1, 3, 4, 5, 6, 7}

(Q’ U R)’                              = {2}

P U (Q’ U R)’                      = {1, 2, 4, 6} U {2}

                                             = {1, 2, 4, 6}

(P U Q) ∩ (P’ ∩ R)’           = P U (Q’ U R)’

{1, 2, 4, 6}                           = {1, 2, 4, 6}

 JADI TERBUKTI SAMA.