#### Properties of Union on Sets

- A U B = B U A
- A U A= A
- A U Φ = A
- A U U = U
- If A ⊂ B then A U B = B
- ( A U B ) U C = A U (B U C)

#### Properties of Intersection on Sets

- A ∩ B= B ∩ A
- A ∩ A= A
- A ∩ Φ = Φ
- A ∩ U = A
- (A ∩ B) ∩ C= A ∩ (B ∩ C)
- If A ⊂ B then A ∩ B = A

#### Properties of Complement on Sets

- A U A' = U
- A ∩ A' = Φ
- U' = Φ
- Φ ' =U

#### Algebra of sets

The algebra of sets develops and describes the basic properties and laws on sets. The common law on set-theoretical operations of union, intersection, complementation and other relations are mentioned below.

#### Properties on Set Equality and Set Inclusion

Let A, B and C be subsets of a universal set U then

- A = B ⇒ B = A
- A = B, B = C ⇒ A = C
- A ⊂ B , B ⊂ C ⇒ A ⊂ C
- A ⊂ Φ ⇒ A = Φ

#### Laws of algebra on sets

Let A, B and C be subsets of a universal set U then

Laws | Identities over union | Identities over Intersection |

Idempotent Laws | A U A = A | A ∩ A = A |

Identities Laws | A U U = U and A U Φ = A | A ∩ U = A and A ∩ Φ = Φ |

Complement Laws | AU A' = U and (A')'=A | A ∩ A' = Φ and U'=Φ |

Commutative Laws | A U B = B U A | A ∩ B = B ∩ A |

Associative Laws | ( A U B )U C = AU (BU A) | (A∩ B)∩ C=A∩ (B∩ A) |

Distributive Laws | AU (B∩ C)=( A U B )∩ (AU C) | A∩ (BU C)=(A∩ B)U (A∩ C) |

De-Morgan’s Laws | (AU B)'=A' ∩ B' | (AU B)'=A' ∩ B' |

#### Theorem 1

Prove that (AUB)'=A'∩B'

#### Set Builder Method

We start by LHS, then

(AUB)' | =U-(AUB) |

={x:x∈U and x∉(AUB)} | |

={x:x∈U and (x∉ A and x∉B)} | |

={x: (x∈U and x∉ A) and (x∈U and x∉B)} | |

={x:x∈A' and x∈B'} | |

=A'∩B' |

Thus

(AUB)'=A'∩B'

#### Membership Tabular Method

A | B | AUB | (AUB)' | A' | B' | A'∩B' |

1 | 1 | 1 | 0 | 0 | 0 | 0 |

1 | 0 | 1 | 0 | 0 | 1 | 0 |

0 | 1 | 1 | 0 | 1 | 0 | 0 |

0 | 0 | 0 | 1 | 1 | 1 | 1 |

Thus

(AUB)'=A'∩B'

#### Venn-Diagram Method

the shaded region is (AUB)'

the shaded region is B'

the shaded region is B'

Thus

(AUB)'=A'∩B'

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