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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\(|A^{10}|=|A|^{10}=2^{10}=1024\)