Bourbaki

Proposition 6

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

Theory of Sets, chapter II, § 4. Union and intersection of a family of sets

Let ff be a mapping of A into B. For every subset Y of B, we have

f−1⟨B−Y⟩=f−1⟨B⟩−f−1⟨Y⟩.\overset{-1}{f}\langle \mathrm{B} - \mathrm{Y} \rangle = \overset{-1}{f}\langle \mathrm{B} \rangle - \overset{-1}{f}\langle \mathrm{Y} \rangle.

For xx belongs to f−1⟨B−Y⟩\overset{-1}{f}\langle \mathrm{B} - \mathrm{Y} \rangle if and only if f(x)f(x) belongs to B but not to Y, i.e., if and only if xx belongs to f−1 ⟨B⟩\overset{-1}{f}\ \langle \mathrm{B} \rangle but not to f−1 ⟨Y⟩\overset{-1}{f}\ \langle \mathrm{Y} \rangle.

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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Cites

Nothing.

Transcribed from the printed edition, 2004, Springer. Not translated.