Proposition 5

Tag 03R1, permanent. Cite it as [Bourbaki, Tag 03R1].

Theory of Sets, chapter III, § 3. Equipotent sets. Cardinals

(a) Let $(\mathfrak{a}_\iota)_{\iota \in \mathrm{I}}$ be a family of cardinals, and let f be a bijection of a set K onto the index set I. Then

\sum_{x \in \mathrm{K}} \mathfrak{a}_{f(x)} = \sum_{\iota \in \mathrm{I}} \mathfrak{a}_\iota, \qquad \mathop{\mathrm{P}}_{x \in \mathrm{K}} \mathfrak{a}_{f(x)} = \mathop{\mathrm{P}}_{\iota \in \mathrm{I}} \mathfrak{a}_\iota.

(b) Let $(\mathfrak{a}_\iota)_{\iota \in \mathrm{I}}$ be a family of cardinals and let $(\mathrm{J}_\lambda)_{\lambda \in \mathrm{L}}$ be a partition of I. Then

\sum_{\iota \in \mathrm{I}} \mathfrak{a}_\iota = \sum_{\lambda \in \mathrm{L}} \left( \sum_{\iota \in \mathrm{J}_\lambda} \mathfrak{a}_\iota \right), \qquad \mathop{\mathrm{P}}_{\iota \in \mathrm{I}} \mathfrak{a}_\iota = \mathop{\mathrm{P}}_{\lambda \in \mathrm{L}} \left( \mathop{\mathrm{P}}_{\iota \in \mathrm{J}_\lambda} \mathfrak{a}_\iota \right)

("associativity of the sum and product").

(c) Let $((\mathfrak{a}_{\lambda\iota})_{\iota \in \mathrm{J}_\lambda})_{\lambda \in \mathrm{L}}$ be a family (indexed by L) of families of cardinals, and let $\mathrm{I} = \coprod_{\lambda \in \mathrm{L}} \mathrm{J}_\lambda$ . Then

\mathop{\mathrm{P}}_{\lambda \in \mathrm{L}} \left( \sum_{\iota \in \mathrm{J}_\lambda} \mathfrak{a}_{\lambda\iota} \right) \sum_{f \in \mathrm{I}} = \left( \mathop{\mathrm{P}}_{\lambda \in \mathrm{L}} \mathfrak{a}_{\lambda, f(\lambda)} \right)

("distributivity of product over sum").

The relations in (a) follow from the analogous formulae for the union and product of sets, for the fact that $f$ is a bijection implies that if $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ is a family of mutually disjoint sets, the elements of the family $(\mathrm{X}_{f(x)})_{x \in \mathrm{K}}$ are also mutually disjoint (cf. Chapter II, § 4, no. 1, Proposition 1 and § 5, no. 3, Proposition 4).

¶ The relations in (b) are immediate consequences of the associativity formulae for unions and products (Chapter II, § 4, no. 2, Proposition 2 and § 5, no. 5, Proposition 7) and the distributivity of intersection over union (Chapter II, § 5, no. 6, Proposition 8), which shows that if $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ is a family of mutually disjoint sets, then the elements of the family

\left( \bigcup_{\iota \in \mathrm{J}_\lambda} \mathrm{X}_\iota \right)_{\lambda \in \mathrm{L}}

are also mutually disjoint.

¶ Finally, (c) follows from the distributivity of the product over union and intersection (Chapter II, § 5, no. 6, Proposition 9 and Corollary 1).

¶ Let $\mathfrak{a}$ and $\mathfrak{b}$ be two cardinals. If I is a set consisting of two distinct elements (e.g., the cardinal 2), there exists a mapping $f$ of I onto $\{\mathfrak{a}, \mathfrak{b}\}$ which defines a family of cardinals. The sum and product of this family depend only on $\mathfrak{a}$ and $\mathfrak{b}$ (by reason of Proposition 5(a)); these cardinals are called respectively the sum and the product of $\mathfrak{a}$ and $\mathfrak{b}$ , and are denoted by $\mathfrak{a} + \mathfrak{b}$ and $\mathfrak{a}\mathfrak{b}$ . Similarly for the sum and product of three or more cardinals. Proposition 5 then implies the following corollary :

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.

Proposition 5

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