algebra of sets

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The set of all subsets of a universal set E is closed under the binary operations∪(union) and∩(intersection) and the unary operation ′ (complementation). The following are some of the properties, or laws, that hold for subsets A, B and C of E:(i)A∪(BC)=(AB)∪C and A∩(BC)=(AB)∩C, the associative properties.(ii)AB=BA and AB=BA, the commutative properties.(iii)A∪Ø=A and A∩Ø=Ø, where Ø is the empty set.(iv)AE=E and AE=A.(v)AA=A and AA=A.(vi)A∩(BC)=(AB)∪(AC) and A∪(BC)=(AB)∩(AC), the distributive properties.(vii)AA′=E and AA′=Ø.(viii)E′=Ø and Ø′=E.(ix) (A′)′=A.(x) (AB)′=A′∩B′ and (AB)′=A′∪B′, De Morgan's laws.The application of these laws to subsets of E is known as the algebra of sets. Despite some similarities with the algebra of numbers, there are important and striking differences.

(i)A∪(BC)=(AB)∪C and A∩(BC)=(AB)∩C, the associative properties.

(ii)AB=BA and AB=BA, the commutative properties.

(iii)A∪Ø=A and A∩Ø=Ø, where Ø is the empty set.

(iv)AE=E and AE=A.

(v)AA=A and AA=A.

(vi)A∩(BC)=(AB)∪(AC) and A∪(BC)=(AB)∩(AC), the distributive properties.

(vii)AA′=E and AA′=Ø.

(viii)E′=Ø and Ø′=E.

(ix) (A′)′=A.

(x) (AB)′=A′∩B′ and (AB)′=A′∪B′, De Morgan's laws.

Subjects: Mathematics.

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