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Formulas for Set Theory

Notations used in set theory:

  • n(A) – Cardinal number of set A.
  • n(A) – Cardinality of set A.
  • Important Formulas: Sets | Quantitative for GMAT = Ac – complement of set A.
  • U – Universal
  • A ⊂ B – Set A is proper subset of subset of B.
  • A ⊆ B – Set A is subset of set B.
  • ∅ – Null set.
  • a ∈ A – element “a” belongs to set A.
  • A ∪ B – union of set A and set B.
  • A ∩ B – intersection of set A and set B.

Formulas

  1. If A and B are overlapping set, n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
  2. If A and B are disjoint set, n(A ∪ B) = n(A) + n(B).
  3. n(A) = n(A ∪ B) + n(A ∩ B) – n(B).
  4. n(A ∩ B) = n(A) + n(B) – n(A ∪  B).
  5. n(B) = n(A ∪ B) + n(A ∩ B) – n(A).
  6. n(U) = n(A) + n(B) – n(A ∩ B) + n((A ∪ B)c).
  7. n((A ∪ B)c) = n(U) + n(A ∩ B) – n(A) – n(B).
  8. n(A ∪ B) = n(A – B) + n(B – A) + n(A ∩ B).
  9. n(A – B) = n(A ∪  B) – n(B).
  10. n(A – B) = n(A) – n(A ∩ B).
  11. n(Ac) = n(U)- n(A)

Example: If A = { 1, 4, 5, 8} and B = { 6, 8, 10, 12}. Find A ∩ B.
Answer:
A∩B= {8}
Important Formulas: Sets | Quantitative for GMAT

Basic Formula for the Venn Diagram

  • Some basic formulas for Venn diagrams of two and three elements.
  • n ( A ∪ B) = n(A ) + n ( B ) – n ( A∩ B)
  • n (A ∪ B ∪ C) = n(A ) + n ( B ) + n (C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) + n (A ∩ B ∩ C)
  • And so on, where n( A) = number of elements in set A.
  • After understanding the concept the of venn diagram with diagram, we don’t have to remember the formulas.

Venn Diagram for 2 sets

n ( A ∪ B) = n(A ) + n ( B ) – n ( A∩ B)
Where;
X = number of elements that belong to set A only
Y = number of elements that belong to set B only
Z = number of elements that belong to set A and B both (A ∩ B)
W = number of elements that belong to none of the sets A or B
From the above figure, it is clear that
n(A) = x + z ;
n (B) = y + z ;
n(A ∩ B) = z;
n ( A ∪ B) = x +y+ z.
Total number of elements = x + y + z + w

Important Formulas: Sets | Quantitative for GMAT

Venn Diagram for 3 sets

n (A ∪ B ∪ C) = n(A ) + n ( B ) + n (C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) + n (A ∩ B ∩ C)
Where, W = number of elements that belong to none of the sets A, B or C

Example: In a Venn diagram representing animals, if Set A represents mammals and Set B represents birds, what does the overlapping region between A and B signify?
Answer: The overlapping region between Set A (mammals) and Set B (birds) signifies the animals that are both mammals and birds. In the context of real animals, this region would be empty since mammals and birds are distinct categories.
Important Formulas: Sets | Quantitative for GMAT

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