Sets Formula

Sets Formula

A set can be defined as a collection of items that are well-defined and have distinct members. Knowledge of sets aids in the application of set formulae in statistics, geometry, probability and sequencing.

The union, complement, intersection and difference of sets are among the set formulae. Formulas of sets are as follows:

• n(A) as well as  n(B) indicate the total elements within two finite sets B and A respectively, then n(AB) = n(A) + n(B) – n(AB) for any two overlapping sets B and A.

• If n(AB) = n(A) + n(B) = n(A) + n(B) = n(A) + n(B) = n(A) + n(B) = n(A) + (B)

• If A, B, as well as C are three finite sets in U, n(ABC)= n(A) + n(C) +n(B)  – n(BC) – n (A B)- n (A C) + n(ABC)

Formulas of Properties of Sets

Commutativity:

• A⋂B = B⋂A

• A∪B = B∪A

Associativity:

• A⋂ (B⋂C) = (A⋂B)⋂C

• A∪ (B∪C) = (A∪B)∪C

Distributivity: A⋂(B∪C) = (A ⋂B) ∪ (A⋂C)

Idempotent Law:

• A ⋂ A = A

• A ∪ A = A

Law of Ø and ∪:

•  A⋂ Ø = Ø

• U ⋂ A = A

• A ∪ Ø = A

• U ∪ A = U

Sets Formulas for Complementary Sets

• Complement Law : A∪A’ = U, A⋂A’ = Ø and A’ = U –  A

• De Morgan’s Laws: (A ∪B)’ = A’ ⋂B’ and (A⋂B)’ = A’ ∪ B’

• Law of Double complementation: (A’)’ = A

• Laws of Empty set and Universal Set: Ø’ = ∪ and ∪’ = Ø

Sets Formulas for Difference of Sets

• A – A = Ø

• B – A = B⋂ A’

• B – A = B – (A⋂B)

• (A – B) = A if A⋂B =  Ø

• (A – B) ⋂ C = (A⋂ C) – (B⋂C)

• A ΔB = (A-B) U (B- A)

• n(AUB) = n(A – B) + n(B – A) + n(A⋂B)

• n(A – B) =  n(A∪B) – n(B)

• n(A – B) = n(A) – n(A⋂B)

• n(A’) = n(∪) – n(A)

Other Formulas of Sets

• n(U) =  n(A) + n(B) + – n(A⋂B) + n((A∪B)’)

n((A∪B)’) = n(U) +  n(A⋂B) – n(A) – n(B)

Important Formulas: