Proposition 7

Tag 03M3, permanent. Cite it as [Bourbaki, Tag 03M3].

Theory of Sets, chapter III, § 5. Properties of integers

For each pair of subsets A, B of a non-empty set E, we have

(1)

\varphi_{\mathrm{E-A}}(x) = 1 - \varphi_{\mathrm{A}}(x),

(2)

\varphi_{\mathrm{A} \cap \mathrm{B}}(x) = \varphi_{\mathrm{A}}(x)\varphi_{\mathrm{B}}(x),

(3)

\varphi_{\mathrm{A} \cup \mathrm{B}}(x) + \varphi_{\mathrm{A} \cap \mathrm{B}}(x) = \varphi_{\mathrm{A}}(x) + \varphi_{\mathrm{B}}(x)

for all $x \in \mathrm{E}$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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