Number of Elements in a Set

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For Two Sets:

Let:

• n(A): Number of elements in Set A

• n(B): Number of elements in Set B

• n(A∪B): Number of elements in the union of A and B

• n(A∩B): Number of elements common to both A and B

Venn Regions:

Region Set Expression Meaning Formula
A1 A ∩ B′  Only A n(A) − n(A ∩ B)
A2 A ∩ B A and B n(A ∩ B)
A3 B ∩ A′ Only B n(B) − n(A ∩ B)
A4 A′ ∩ B′ Neither A nor B n(Ω) − n(A ∪ B)

Union Formula:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B) 

• If A and B are disjoint: n(A ∩ B) = 0  

⇒ n(A ∪ B) = n(A) + n(B) 

For Three Sets:

Let the sets be A, B, and C.

Venn Regions:

RegionSet ExpressionMeaningFormula
A1A ∩ B′ ∩ C′Only An(A) − n(A ∩ B) − n(A ∩ C) + n(A ∩ B ∩ C)
A2B ∩ A′ ∩ C′Only Bn(B) − n(A ∩ B) − n(B ∩ C) + n(A ∩ B ∩C)
A3C ∩ A′ ∩ B′Only Cn(C) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)
A4 A ∩ C ∩ B′ A and C only n(A ∩ C) −  n(A ∩ B ∩ C)
A5  A ∩ B ∩ C′ A and B only n (A ∩ B) −  n(A ∩ B ∩ C)
A6 B ∩ C ∩ A′  B and C only n(B ∩ C) −  n(A ∩ B ∩ C)
A7 A ∩ B ∩ CAll three n(A ∩ B ∩ C)
A8 A′ ∩ B′ ∩ C′None of them n(Ω) − n(A ∪ B ∪ C)
             

Union Formula:

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n (A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∪ B ∪ C)


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