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Proposition: Indicator Function and Set Operations

Let $S$ be a set, $A,B\subseteq S$ be subsets with the indicator functions $\chi_A$ and $\chi_B.$ Then the equations hold:

• for the set union: $\chi_{A\cup B}=\max(\chi_A,\chi_B)=\chi_A+\chi_B-\chi_{A\cap B},$
• for the set intersection: $\chi_{A\cap B}=\min(\chi_A,\chi_B),$
• for the set complement: $\chi_{A^C}=\chi_S-\chi_A,$
• for the set difference: $\chi_{A\setminus B}=\chi_A-\chi_A\cdot\chi_B,$

Table of Contents

Proofs: 1

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References

Bibliography

1. Flachsmeyer, Jürgen: "Kombinatorik", VEB Deutscher Verlag der Wissenschaften, 1972, 3rd Edition