Set Theory formulas

Set Theory is a branch of mathematics which deals with the study of sets or the collection of similar objects.

Set theory is one of the most fundamental branch of mathematics, But is also also very complex if you try to analyze three or more sets.

You learn some important set theory formulas in this page which helps you to analyze the group of three or less sets.

If you imagine three sets as:



Note: The sets are overlapping just for the sake of ease, The formulas given here also implies to sets if the overlapped part is null.

Then the following formulas should be correct in the situation:

Set Theory Formulas:

Notations used in set theory formulas:

n(A) – Cardinal number of set A.

n_o(A) – cardinality of set A.

\bar{A} = A^c – Complement of set A.

U – universal set

A \subset B – Set A is proper subset of subset of set B.

A \subseteq B – Set A is subset of set B.

 \phi – Null set.

 a \in A – element “a” belongs to set A.

 A \cup B – Union of set A and set B.

 A \cap B – Intersection of set A and set B.


For a group of two sets:

1. If A and B are overlapping set, n(A \cup B) = n(A) + n(B) - n(A \cap B)

2. If A and B are disjoint set, n(A \cup B) = n(A) + n(B)

3. n(A) = n(A \cup B) +n(A \cap B) - n(B)

4. n(A \cap B) = n(A) +n(B) - n(A \cup B)

5. n(B) = n(A \cup B) +n(A \cap B) - n(A)

6. n(U) = n(A) + n(B) - n(A \cap B) + n((A \cup B)^c)

7. n((A \cup B)^c) = n(U) + n(A \cap B) - n(A) - n(B)

8. n(A \cup B) = n(A - B) + n(B - A) + n(A \cap B)

9. n(A - B) = n(A \cup B) - n(B)

10. n(A - B) = n(A) - n(A \cap B)

11. n(A^c) = n(U) - n(A)

For a group of three sets:
1. n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C

2. n(A \cup B \cup C) = n(U) - n((A \cup B \cup C)^c)

3. n(A \cap B \cap C) = n(A \cup B \cup C)+n(A \cap B)+n(A \cap C)+n(B \cap C) - n(A) - n(B) - n(C)

4.  n_0(A) = n(A)-n(A \cap B)-n(A \cap C)+n(A \cap B \cap C)

5.  n_0(B) = n(B)-n(A \cap B)-n(B \cap C)+n(A \cap B \cap C)

6.  n_0(C) = n(C)-n(A \cap C)-n(B \cap C)+n(A \cap B \cap C)

7. n(A \cap B only) = n(A \cap B)-n(A \cap B \cap C)

8. n(B \cap C only) = n(B \cap C)-n(A \cap B \cap C)

9. n(A \cap C only) = n(A \cap C)-n(A \cap B \cap C)

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