Exercise 3, § 3

Tag 03L5, permanent. Cite it as [Bourbaki, Tag 03L5].

Theory of Sets, III, 229

Bourbaki marks this one as harder than the rest.

Let $(\mathfrak{a}_\iota)_{\iota \in \mathrm{I}}$ and $(\mathfrak{b}_\iota)_{\iota \in \mathrm{I}}$ be two families of cardinals such that $\mathfrak{b}_\iota \geqslant 2$ for each $\iota \in \mathrm{I}$ .

(a) Show that. if $\mathfrak{a}_\iota \leqslant \mathfrak{b}_\iota$ for each $\iota \in \mathrm{I}$ , then

\sum_{\iota \in \mathrm{I}} \mathfrak{a}_\iota \leqslant \mathbf{P}_{\iota \in \mathrm{I}} \mathfrak{b}_\iota.

(b) Show that, if $\mathfrak{a}_\iota < \mathfrak{b}_\iota$ for each $\iota \in \mathrm{I}$ , then

\sum_{\iota \in \mathrm{I}} \mathfrak{a}_\iota < \mathbf{P}_{\iota \in \mathrm{I}} \mathfrak{b}_\iota.

(Note that a product $\prod_{\iota \in \mathrm{I}} \mathrm{E}_\iota$ cannot be the union of a family $(\mathrm{A}_\iota)_{\iota \in \mathrm{I}}$ such that $\mathrm{Card}\,(\mathrm{A}_\iota) < \mathrm{Card}\,(\mathrm{E}_\iota)$ for all $\iota \in \mathrm{I}$ , by observing that $\mathrm{Card}\,(\mathrm{pr}_\iota(\mathrm{A}_\iota)) < \mathrm{Card}\,(\mathrm{E}_\iota).$ )

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