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 \( (\emptyset \cap A) = \emptyset \)
  • \( (A - \emptyset) = A \)

• \( (A \space \triangle \space A) = \emptyset \)