題目
定義
Cartisan Product (ordered pair)
\(
A\times B
\)
Definition
\(
A\times B
=
\{
(a,b)|a\in A, b\in B
\}
\)
\(
(a,b)
\) is a ordered pair.
Suppose
\(
a\neq b
\)
\(
(a,b) \neq (b,a)
\)
Suppose
\(
|A|=m,
|B|=n
\),
\(
|A\times B|=mn
\)
example
\(
A=\{1,2,3\}
\),
\(
B=\{4,5\}
\)
\(
A\times B=
\{
(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)
\}
\)
\(
|A|=3, |B|=2
\)
\(
|A\times B|=6
\)
Excerise
Suppose
\(
A=\{0.1,0.2\},
B=\{0.3\}
\),
\(
\text{Determine }2^B \text{ and } A\times 2^B\text{?}
\)
solution
\(
2^B=
\{\emptyset, \{0.3\}\}
\)
\(
A\times 2^B=
\)
\(
\{\;(0.1,\emptyset),\;(0.1,\{0.3\}),\;(0.2,\emptyset),\;(0.2,\{0.3\})\;\}
\)
Excerise
Let \( A \) be the set \( \{1,2,3\} \) and
\( B \) be the set \( \{3.14, 2.71\} \).
How many elements are there in the set
\(
2^{2^A\times 2^B}
\)?
solution
ans.
\(
|2^{2^A\times 2^B}|
\)
\(
|A|=3, |B|=2
\)
\(
|2^A|=|\mathcal{P}(A)|=2^3,
|2^B|=|\mathcal{P}(B)|=2^2,
\)
\(
|2^A\times 2^B|=2^3\times 2^2=2^5
\)
\(
|2^{2^A\times 2^B}|=|\mathcal{P}(2^A\times 2^B)|
=
2^{2^5}
=2^{32}
\)