Important Formulas: Sets | Quantitative for GMAT PDF Download

Notations used in set theory:

• n(A) – Cardinal number of set A.
• n∘(A) – Cardinality of set A.
• = A^c – 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}

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.