Bourbaki

Exercise 4, § 4

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

Theory of Sets, II, 126

Let (Gι)ι∈I(\mathrm{G}_\iota)_{\iota \in \mathrm{I}} be a family of graphs. Show that for every set X we have

(⋃ι∈IGι)⟨X⟩=⋃ι∈IGι⟨X⟩,\left( \bigcup_{\iota \in \mathrm{I}} \mathrm{G}_\iota \right) \langle \mathrm{X} \rangle = \bigcup_{\iota \in \mathrm{I}} \mathrm{G}_\iota \langle \mathrm{X} \rangle,

and that for every object xx,

(⋂ι∈IGι)⟨{x}⟩=⋂ι∈IGι⟨{x}⟩.\left( \bigcap_{\iota \in \mathrm{I}} \mathrm{G}_\iota \right) \langle \{x\} \rangle = \bigcap_{\iota \in \mathrm{I}} \mathrm{G}_\iota \langle \{x\} \rangle.

Give an example of two graphs G, H and a set X such that

(G∩H)⟨X⟩≠G⟨X⟩∩H⟨X⟩.(\mathrm{G} \cap \mathrm{H}) \langle \mathrm{X} \rangle \neq \mathrm{G} \langle \mathrm{X} \rangle \cap \mathrm{H} \langle \mathrm{X} \rangle.
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Transcribed from the printed volume. Not translated.