Sets Formulas-About, Definition, Use & Solved Examples

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About Sets Formulas

Set formulas are formulas associated with set theory in maths. Set is a collection of well-defined objects and it has different elements. Application of set formulas, in areas related to stat, probability, geometry and sequence.

Set formulas include union, intersection, complement and difference of sets. The Venn diagram is popularly applied to visualize set formulas to arrive at their proof.

What Are Sets Formulas?

Set formulas have been derived from set theory, which can be applied for ready reference. Now take a look at set notation, symbols, definitions, and properties of sets before the formula.

If n(A) and n(B) denote number of elements in two finite set A and B respectively, hence for any two overlapping sets A and B,

n (A $\cup$ B) = n (A) + n(B) - n(A $\cup$ B)
If A and B are disjoint sets,n (A $\cup$ B) = n (A) + n(B)
If A, B and C are 3 finite sets in U then, n (A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C) - n(B $\cap$ C) - n(A $\cap$ B) - n(A $\cap$ C) + n(A $\cap$ B $\cap$ C)

Some Important Formulae: For any three sets A, B, C.
S. No.	Formulae
1	n(A $\cup$ B) = n (A) + n(B) - n(A $\cup$ B)
2	If (A $\cap$ B) = φ, then n (A $\cup$ B) = n (A) + n(B)
3	n(A - B) + n(A $\cap$ B) = n(A)
4	n(B - A) + n(A $\cap$ B) = n(B)
5	n(A $\cup$ B) = n(A - B) + n(A $\cap$ B) + n(B - A)
6	n (A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C) - n(A $\cap$ B) - n(B $\cap$ C) - n(A $\cap$ C) + n(A $\cap$ B $\cap$ C)

Sets Formulas on Properties of Sets

Set formulas have almost similar properties as real numbers or natural numbers. Sets also follow the commutative property, associative property, and distributive property. Set formula based on the properties of sets is as follows.

Commutativity:

A $\cap$ B = B $\cap$ A
A $\cup$ B = B $\cup$ A

Associativity:

A $\cap$ (B $\cap$ C) = (A $\cap$ B) $\cap$ C
A $\cup$ (B $\cup$ C) = (A $\cup$ B) $\cup$ C

Distributivity::

Idempotent Law:
A ∩ A = A
A ∪ A = A
Law of φ and ∪:
A ∩ φ = φ
U ∩ A = A
A ∪ φ = A
U ∪ A = U
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Sets Formulas of Complement Sets

Set formulas for complement for a set include basic complement law, de morgan's laws, double complement, and law of empty set and universal set

Complement Law: A ∪ A' = U, A ∩ A' = φ 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: φ = ∪ and ∪' = φ

Sets Formulas of Difference of Sets

Set formulas or difference of sets across two sets, across a null set and for the complement of a set is as follows.

A - A = φ
B - A = B ∩ A'
B - A = B - (A ∩ B)
(A - B) = A if A ∩ B = φ
(A - B) ∩ C = (A ∩ C) - (B ∩ C)
A Δ B = (A - B) ∪ (B - A)
n(A ∪ B) = 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 Important Sets Formulas

n(∪) = n(A) + n(B) ± n(A ∩ B) + n((A ∪ B)')
n((A ∪ B)') = n(∪) + n(A ∩ B) - n(A) - n(B)

Download all the Maths formulas from the HT maths page.

Sets Formulas