Set operations formula#
Union of Two Sets (cardinality)#
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
how to read:
whole A and B add together, its make a data duplication on center of venn diagram, we remove that duplication using \( |A \cap B| \)
Intersection (steps)#
its flip of Union \[ |A \cup B| = |A| + |B| - |A \cap B| \] \[ |A \cap B| = |A| + |B| - |A \cup B| \]
Intersection of three sets#
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \]
how it works, de remove data duplication two times on \( |A \cap B| \), \( |B \cap C| \), and \( |B \cap C| \), then fill the empty section on the center which \( |A \cap B \cap C| \)
This is proof of concept by nice guy on internet: https://www.youtube.com/watch?v=vVZwe3TCJT8.
Difference#
\[ |A - B| = |A| - |A \cup B| \]
Symetric difference#
1. General formula#
\[ A \space \triangle \space B = (A \cup B) - (A \cap B) \] Cardinals version: \[ | A \space \triangle \space B | = |A - B| + |B - A| \] \[ | A \space \triangle \space B | = |A| + |B| - 2|A \cap B| \]
2. Symetric difference properties#
\( A \space \triangle \space B = B \space \triangle \space A \)
\( (A \space \triangle \space B) \space \triangle C = A \space \triangle \space (B \space \triangle C) \)
\( (A \space \triangle \space \emptyset) = A \), why?
\(A \space \triangle \space B = (A \cup B) - (B \cap A) \)
\(A \space \triangle \space \emptyset = (A \cup \emptyset) - (\emptyset \cap A) \), but
\( (A \cup \emptyset) = A \) and \( (\emptyset \cap A) = \emptyset \)
\( (A - \emptyset) = A \)
\( (A \space \triangle \space A) = \emptyset \)