Products of sets

Section 5.11 Products of sets

An indexed set is really just another name for a map

U : A \to Ξ;

we typically abuse notation and write

U_{a}

instead of

U (a),

and we write

(U_{a})_{a \in A}

for the map

U .

The union of the indexed set

(U_{a})_{a \in A}

will mean the union of the set

{U_{a} : a \in A} :

⋃_{a \in A} U_{a} := ⋃ {U_{a} : a \in A} .

🔗

It may seem a bit silly to belabour this point, but the purpose will, we hope, become clear.

🔗

For any indexed set

(U_{a})_{a \in A},

the product is the set

\prod_{a \in A} U_{a} := {x \in Map (A, ⋃_{a \in A} U_{a}) : (\forall a \in A) x (a) \in U_{a}} .

🔗

An element of

\prod_{a \in A} U_{a}

is thus a map

x : A \to ⋃_{a \in A} U_{a}

such that for every

a \in A,

one has

x (a) \in U_{a} .

One usually writes

x_{a}

instead of

x (a),

and one often writes

x = (x_{a})_{a \in A} .

🔗

Example 5.11.1.

When \(A = \{1,2\}\text{,}\) an indexed set consists of two sets \(U_1\) and \(U_2\text{.}\) The product \(\prod_{a \in \{1,2\}} U_a\) thus consists of pairs \((x_1, x_2)\) with \(x_1 \in U_1\) and \(x_2 \in U_2\text{.}\) The assignment \((x_1,x_2) \mapsto \angs{x_1,x_2}\) is a bijection \(\prod_{a \in \{1,2\}} U_a \to U_1 \times U_2\text{.}\) Most mathematicians are happy to pretend as if there is no difference between these sets; indeed, there is no interesting difference!

🔗

More generally, we think of the product

\prod_{a \in A} U_{a}

as the set of ordered “

A

-tuples”.

🔗

For every

b \in A,

there is an attached map

π_{b} : \prod_{a \in A} U_{a} \to U_{b}

🔗

given by the assignment

x \mapsto x_{b},

called the projection onto the

b

-th factor.

🔗

For every set

S,

every indexed set

{U_{a}}_{a \in A},

and every indexed set

{f_{a} : S \to U_{a}}

of maps, there exists a unique map

f : S \to \prod_{a \in A} U_{a}

🔗

such that

π_{a} \circ f = f_{a} .

Indeed, the map

f

is given by the assignment

s \mapsto (f_{a} (x))_{a \in A} .