KALKULUS HIMPUNAN

SOAL

Buktikan jika

(P ᴗ Q) ᴖ (P' ᴖ R)' = P ᴗ (Q' ᴗ R)'

Misalkan :

S = {1,2,3,4,5,6,7,8,9,10}

P = {1,3,5,7,9}

Q = {2,4,6,8,10}

R = {2,3,6,7,8}

PEMBUKTIAN

Jawab :

P ᴗ Q = {1,2,3,4,5,6,7,8,9,10}

 P' = {2,4,6,8,10}

P' ᴖ R = {2,4,6,8,10} ᴖ {2,3,6,7,8} = {2,6,8}

(P' ᴖ R)' = {1,3,4,5,7,9,10}

(P ᴗ Q) ᴖ (P' ᴖ R)' = {1,2,3,4,5,6,7,8,9,10} ᴖ {1,3,4,5,7,9,10} = {1,3,4,5,7,9,10}

 Q' = {1,3,5,7,9}

Q' ᴗ R = {1,3,5,7,9} ᴗ {2,3,6,7,8} = {1,2,3,5,6,7,8,9}

(Q' ᴗ R)' = {4,10}

P ᴗ (Q' ᴗ R)' = {1,3,5,7,9} ᴗ [4,10} = {1,3,4,5,7,9,10}

Jadi benar bahwa (P ᴗ Q) ᴖ (P' ᴖ R)' = P ᴗ (Q' ᴗ R)'