Sifat-sifat Himmpunan

Sifat Komutatif

A={1,3,5}A=\{1,3,5\}B={2,3,6}B=\{2,3,6\}

Pada Irisan Himpunan

A∩B={3}A\cap B=\{3\}B∩A={3}B\cap A=\{3\}

∴A∩B=B∩A\therefore A\cap B=B\cap A

Pada Gabungan Himpunan

A∩B={1,2,3,5,6}A\cap B=\{1,2,3,5,6\}B∩A={1,2,3,5,6}B\cap A=\{1,2,3,5,6\}

∴A∪B=B∪A\therefore A\cup B=B\cup A

Sifat Asosiatif

A={1,2,3,4,5}A=\{1,2,3,4,5\}B={4,5,6,7,8,9}B=\{4,5,6,7,8,9\}C={3,5,9,10,11,12}C=\{3,5,9,10,11,12\}

Pada Irisan Himpunan

A∩B={4,5}A\cap B=\{4,5\}B∩C={5,9}B\cap C=\{5,9\}A∩(B∩C)={5}A\cap (B\cap C)=\{5\}(A∩B)∩C={5}(A\cap B)\cap C=\{5\}

∴A∩(B∩C)=(A∩B)∩C\therefore A\cap (B\cap C)=(A\cap B)\cap C

Pada Gabungan Himpunan

A∪B={1,2,3,4,5,6,7,8,9}A\cup B=\{1,2,3,4,5,6,7,8,9\}B∪C={3,4,5,6,7,8,9,10,11,12}B\cup C=\{3,4,5,6,7,8,9,10,11,12\}A∪(B∪C)={1,2,3,4,5,6,7,8,9,10,11,12}A\cup (B\cup C)=\{1,2,3,4,5,6,7,8,9,10,11,12\}(A∪B)∪C={1,2,3,4,5,6,7,8,9,10,11,12}(A\cup B)\cup C=\{1,2,3,4,5,6,7,8,9,10,11,12\}

∴A∪(B∪C)=(A∪B)∪C\therefore A\cup (B\cup C)=(A\cup B)\cup C

Sifat Distributif

A={1,2,3,4,5}A=\{1,2,3,4,5\}B={4,5,6,7,8,9}B=\{4,5,6,7,8,9\}C={3,5,9,10,11,12}C=\{3,5,9,10,11,12\}

Pada Irisan Himpunan terhadap Gabungan Himpunan

A∩B={4,5}A\cap B=\{4,5\}A∩C={3,5}A\cap C=\{3,5\}B∪C={3,4,5,6,7,8,9,10,11,12}B\cup C=\{3,4,5,6,7,8,9,10,11,12\}A∩(B∪C)={3,4,5}A\cap (B\cup C)=\{3,4,5\}(A∩B)∪(A∩C)={3,4,5}(A\cap B)\cup (A\cap C)=\{3,4,5\}

∴A∩(B∪C)=(A∩B)∪(A∩C)\therefore A\cap (B\cup C)=(A\cap B)\cup (A\cap C)

Pada Gabungan Himpunan terhadap Irisan Himpunan

A∪B={1,2,3,4,5,6,7,8,9}A\cup B=\{1,2,3,4,5,6,7,8,9\}A∪C={1,2,3,4,5,9,10,11,12}A\cup C=\{1,2,3,4,5,9,10,11,12\}B∩C={5,9}B\cap C=\{5,9\}A∪(B∩C)={1,2,3,4,5,9}A\cup (B\cap C)=\{1,2,3,4,5,9\}(A∪B)∩(A∪C)={1,2,3,4,5,9}(A\cup B)\cap (A\cup C)=\{1,2,3,4,5,9\}

∴A∪(B∩C)=(A∪B)∩(A∪C)\therefore A\cup (B\cap C)=(A\cup B)\cap (A\cup C)

Operasi himpunan