Set Identities Formulas

Definitions:

Universal set: I
Empty set: \(\emptyset\)

Union of sets

\(A \cup B = \{x|x \in A  or x \in B\} \)

Intersection of sets

\(A \cap B = \{x|x \in A  and  x \in B\} \)

Complement

\( A’ = \{ x  \in  l | x \in A \} \)

Difference of sets

\( B \backslash A = \{ x | x \in B  and  x \notin A \} \)

Cartesian product

\( A x B = \{ ( x, y ) | x \in A  and  y \in B \} \)

Set identities involving union

Commutativity

\( A \cup B = B \cup A \)

\( A \cup (B \cup A) = (A \cup B) \cup C \)

\( A \cup A = A \)

Set Identities involving intersection

Commutativity

\( A \cap B = B \cap A \)

Associativity

\( A \cap (B \cap C) = (A \cap B) \cap C \)

\( A \cap A = A \)

Set identities involving union and intersection

Distributivity

\( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)

\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)

Domination

\( A \cap \emptyset = \emptyset \)

\( A \cup l = l \)

Identity

\( A \cup \emptyset = A \)

\( A \cap l = A \)

Set identities involving union, intersection and complement

Complement of intersection and union

\( A \cup A’ = l \)

\(A \cap A’ = \emptyset  \)

De Morgan’s laws

\( (A \cup B)’ = A’ \cap B’ \)

\( (A \cap B)’ = A’ \cup B’ \)

Set identities involving difference

\( B \backslash A = B (A \cup B) \)

\(B \backslash A = B \cap A’ \)

\(A \backslash A = \emptyset \)

\( (A \backslash B) \cap C = (A \cap C) \backslash (B \cap C) \)

\( A’ = l \backslash A \)

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