A set is simply a collection of well-defined individual objects that form a group. A set can contain any group of items, such as a set of numbers, a day of the week, or a vehicle.
Example: A = { 2, 4, 6, 8 }.
A is a set and 2, 4, 6, and 8 are elements of the set. Elements written in a set can be used in any order, but cannot be repeated.
Set Formulas
A set formula is a formula related to set theory in mathematics. Set knowledge helps you apply set formulas in areas related to statistics, probability, geometry, and sequences.
Set Formulas on the Number of Elements of Sets
If n(A) and n(B) represent the number of elements in two finite sets A and B, respectively, then
- n (A U B) = n(A) + n(B) - n(A ∩ B)
- n(A) = n(A U B) + n(A ∩ B) - n(B)
- n(B) = n(A U B) + n(A ∩ B) - n(A)
If sets A and B are disjoint sets then
- n(A U B) = n(A) + n(B)
- A ∩ B = ∅
- n(A - B) = n(A)
Sets Formulas on Properties of Sets
Set formulas have almost the same properties as real or natural numbers. Sets also have commutative, associative, and distributive properties. The set formula according to the properties of the set is as follows.
- 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 U
A⋂ Ø = Ø
U ⋂ A = A
A ∪ Ø = A
U ∪ A = U
Sets Formulas of Complement Sets
The set formulas of set complement include the basic law of complement, De Morgan's law, double complement, and the law of empty and universal 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' = U and U' = Ø
Sets Formulas of 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) - n(A⋂B)
- n(A') = n(∪) - n(A)