Rule of product

Statement

Let $A$ and $B$ be finite sets. Then the Cartesian product $A \times B = \{(a, b) : a \in A, b \in B\}$ is finite and $|A \times B| = |A| \cdot |B|$ .

More generally, for finite sets $A_1, \dots, A_k$ , the product $A_1 \times \dots \times A_k$ has $|A_1| \cdots |A_k|$ elements.

Remarks

A common form of the rule: a choice is made in $k$ steps, and step $i$ has $n_i$ options, whatever the earlier steps are. Then there are $n_1 n_2 \cdots n_k$ choices in total. The number of functions between finite sets is a direct consequence.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Cardinality of a finite set definition
Rule of product statement

Where this leads

2 pages build on this one.

Tag	`0VPK4X73Q0`
URI	`m2://0VPK4X73Q0`
Aliases	`multiplication-principle`, `product-rule`
Kind	statement, theorem
Writing	W1 stated
MSC	05A05
Status	This copy of the site has no build data. Run `m2 status 0VPK4X73Q0`.
Uses	Cardinality of a finite set
Same as	Wikidata Q557624 Wikipedia Rule of product

Rule of product

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by