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.

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.

Formulas:

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

Set Theory formulas

Set Theory formulas

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