Ordered pair & triple and Cartesian product of sets

Naive set theory: Elementary notions and notations

Ordered pair & triple and Cartesian product of sets

Ordered pair and Cartesian product

Ordered pair

An ordered pair $(x,y)$ is a pair of objects in which there is a first component $x$ and a second component $y$. Two ordered pairs $(u,v)$ and $(x,y)$ are equal if and only if $u=x$ and $v=y$. In general, $(x,y)$ will not be equal to $(y,x)$. The order of the objects in an ordered pair matters.

Cartesian product

The Cartesian product of the sets $X$ and $Y$, denoted as $X\times Y$, is defined as the set of ordered pairs $(x,y)$ where $x$ belongs to $X$ and $y$ belongs to $Y$. In formula form: \[X\times Y = \{(x,y)\mid x\in X\text{ and } y\in Y\}\] We also denote $X\times X$ as $X^2$.

Examples

If $X=\{1,2\}$ and $Y=\{a,b\}$, then \[\begin{aligned} X\times Y&=\bigl\{(1,a), (1,b), (2,a), (2,b)\bigr\}\\[0.25cm]
Y\times X&=\bigl\{(a,1), (a,2), (b,1), (b,2)\bigr\}\\[0.25cm]
X\times X&=\bigl\{(1,1), (1,2), (2,1), (2,2)\bigr\}\end{aligned}\]
$\mathbb{R}\times \mathbb{R} = \mathbb{R}^2$

Formal definition of ordered pair

We can also formally define an ordered pair $(x,y)$ within set theory; for example, via the Kuratowski definition from 1921: $(x,y)=\bigl\{\{x\}, \{x,y\}\bigr\}$.
Under this definition, it is true that if \(x\in X$ and $y\in Y$, then $(x,y)\in {\Large\wp}\bigl({\Large\wp}(X\cup Y)\bigr)$.

The above definition of an ordered pair is not the only possibility: in 1917, Wiener proposed $(x,y)=\bigl\{\{x\}, \{y,\emptyset\}\bigr\}$.

We could also use $(x,y)=\Bigl\{\{x,\emptyset\}, \bigl\{y,\{\emptyset\}\bigr\}\Bigr\}$ as a definition of an ordered pair.

Ordered triple

Ordered triple

An ordered triple $(x,y,z)$ is a set of three objects in which there is a first component $x$, a second component $y$, and a third component $z$. Two ordered triples $(u,v,w)$ and $(x,y,z)$ are equal if and only if $u=x$ and $v=y$ and $w=z$. The order of the objects in an ordered triple matters.

Cartesian product

The Cartesian product of the sets $X$, $Y$, and $Z$, denoted as $X\times Y\times Z$, is defined as the set of ordered triples $(x,y,z)$ where $x$ belongs to $X$, $y$ belongs to $Y$, and $z$ belongs to $Z$. In formula form: \[X\times Y\times Z= \{(x,y,z)\mid x\in X\text{ and } y\in Y\text{ and } z\in Z\}\] We also denote $X\times X\times X$ as $X^3$.

Examples

If $X=\{1,2\}$, $Y=\{a,b\}$, and $Z=\{\alpha,\beta\}$, then \[\begin{aligned} X\times Y\times Z &=\bigl\{(1,a,\alpha), (1,a,\beta),\\ &\phantom{=\bigl\{\;} (1,b,\alpha), (1,b,\beta),\\ &\phantom{=\bigl\{\;} (2,a,\alpha), (2,a,\beta),\\
&\phantom{=\bigl\{\;} (2,b,\alpha), (2,b,\beta)\bigr\}\end{aligned}\] $\mathbb{R}\times \mathbb{R} \times \mathbb{R}= \mathbb{R}^3$

Formele definitie van geordend tripel

We can also formally define an ordered triple $(x,y,z)$ within set theory; for example by defining a triple as $(x,y,z)=\bigl\{\{x\}, \{x,y\}, \{x,y,z\}\bigr\}$ .

Recursive definition of an ordered triple

We can also recursively define an ordered triple$(x,y,z)$. Because $X\times Y$ is a set, you can also construct $(X\times Y) \times Z$, and so on. Alas, it is not true that $\(X\times (Y \times Z)$. But there does exist an identification between elements of these two sets.

Ordered n-tope

Ordered $\boldsymbol{n}$-tuple

An ordered $\boldsymbol{n}$-tuple $(x_1,x_2,\ldots x_n)$ can be defined recursively for every natural number $n\ge 1$ as \[\begin{aligned} (x_1,x_2,\ldots x_n) &= \bigl((x_1,x_2,\ldots, x_{n-1}),x_n\bigr)\\ (x) &= x\end{aligned}\] For every $n\in \mathbb{N}$, $n\ge 1$: $(x_1,x_2,\ldots, x_n)=(y_1,y_2,\ldots, y_n)$ if and only if $x_1=y_1$ and $x_2=y_2$ and $\dots$ and $x_n=y_n$.

We denote the product of $n$ times the same set $X$ as $X^n$.

Examples

$\mathbb{R}^3=(\mathbb{R}\times \mathbb{R})\times\mathbb{R}$

If $X=\{1,2\}$, then with recursion: \[\begin{aligned} X^3 &=X^2\times X\\[0.25cm]
&=\bigl\{(1,1),(1,2), (2,1), (2,2)\bigr\}\times\{1,2\}\\[0.25cm]
&= \Bigl\{\bigl((1,1),1\bigr),\bigl((1,1),2\bigr),\bigl((1,2),1\bigr)\\[0.25cm] &\phantom{===}\! \bigl((1,2),2\bigr),
\bigl((2,1),1\bigr),\bigl((2,1),2\bigr),\\[0.25cm]&\phantom{===} \!\bigl((2,2),1\bigr),\bigl((2,2),2\bigr)\Bigr\}\end{aligned}\]

For all sets $X$, $Y$, $Z$ and $W$: \[\begin{aligned} X\times(Y\cap Z) &= (X\times Y)\cap (X\times Z)\\[0.25cm] X\times(Y\cup Z) &= (X\times Y)\cup (X\times Z)\\[0.25cm] (X\times Y)\cap (Z\times W) &= (X\cap Z)\times (Y\cap W)\\[0.25cm] (X\times Y)\cup (Z\times W) &\subset (X\cup Z)\times (Y\cup W)\end{aligned}\]