Cartesian Product

The Cartesian product of two sets $S$ and $T$, denoted as $S \times T$, is the set of ordered pairs $(x,y)$ with $x \in S$ and $y \in T$.

In symbols, $S \times T = \{(x,y)|x\in S \wedge y\in T\}$

Let $S=\{1,2,3\},T=\{4,5,6\}$. What is $S\times T$?

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Referring to the definition, $S \times T=\{(x,y)|x\in S \wedge y\in T\}=\{(x,y)|x\in \{1,2,3\} \wedge y\in \{4,5,6\}\}$.

We can list the elements out: $S \times T = \{(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)\}$

Cartesian product of finite number of sets is similar to that of two sets. Instead of a formal definition, an example is given here.

Let $A=\{1,2\}, B=\{3,4\}, C=\{5,6\}$. What is $A\times B\times C$?

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Listing the elements out: $A\times B\times C = \{(1,3,5),(1,3,6),(1,4,5),(1,4,6),(2,3,5),(2,3,6),(2,4,5),(2,4,6)\}$

From product of numbers we can define power of numbers. Similarly, from Cartesian product we can define Cartesian power.

$S^n=\underbrace{S\times S\times \cdots \times S}_{\text{n times}}$

$\mathbb R^3=\{(a,b,c)|a\in \mathbb R,b\in \mathbb R,c\in \mathbb R\}$

which is all 3-tuples with real number as elements.

We have $|A\times B|=|A|\times |B|$ where $|A|$ denotes the cardinality of the set $A$.

Let $A=\{1,2,3,4,5\},B=\{a,b,c,d\}$. What is $|A\times B|$?

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$|A\times B|=|A|\times |B|=5\times 4=20$

We have $|A^n|=|A|^n$

Let $A=\{0,1\}$. What is $|A^{10}|$?

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$|A^{10}|=|A|^{10}=2^{10}=1024$