MATHEMATICS

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Kamis, 05 September 2013

Set Theory formulas

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)
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