Ordered Pair And Cartesian Product | Relations And Functions

Ordered Pair

There are two elements in a pair. $\text{(Earth, Mars)}$, $\text{{Maths, Physics}}$, $(4,5)$, $\text{{a,b}}$ are some examples of pair. Two pairs such as $\text{{Maths, Physics}}$ and $\text{{Physics, Maths}}$ have the same elements. Similarly, $(4, 5)$ and $(5, 4)$ also have the same elements. But in each case, the elements are distinct. If we consider a case in which the pair is $(3, 3)$, then the pair has two identical elements. Two types of brackets are used here;

Curly brackets $\{ \}$ or braces are used to denote sets. In sets, elements do not have the order of their occurence i.e. the elements can occur anywhere within $\{ \}$. But, the repeatation of a same element is not allowed in sets.
If the elements have the order in which they occur and an element can be repeated, then parentheses $( , )$ with a comma between the elements are used. Such pair of elements is said to be ordered.

An ordered pair is a pair having one element as the first and the other as the second. An ordered pair having a as the first element and b as the second element is denoted by $(a, b)$.

An ordered pair $(a, b)$ is not the same as the ordered pair $(b, a)$ unless the two elements are identical. Thus, $(4, 5)$ is different from $(5, 4)$; but $(3, 3)$ is the same as $(3, 3)$. Two ordered pairs $(a, b)$ and $(c, d)$ are equal if and only if $a=c$ and $b=d$. Thus, two ordered pairs $(2, 3)$ and $(2, 3)$ are equal, but the ordered pairs $(2, 3)$ and $(3, 2)$ are not equal.

Cartesian Product

Let $A$ and $B$ be two given sets. Then, the set of all ordered pairs $(a, b)$ such that $a∈A$ and $b∈B$ is called the Cartesian Product of $A$ and $B$, and is denoted by $A×B$ (read as $A$ cross $B$). In the set builder notation, \[A×B=\{(a, b): a∈A \text{ and } b∈B\}\]

Let $A=\{1, 2, 3\}$ and $B=\{4, 5\}$. Then,

$A×B=\{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)\}$

and, $B×A=\{(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)\}$

The Cartesian Product of $A=\{H, T\}$ with itself, also known as the Cartesian Product on $A$ is,

$A×A=\{(H, H), (H, T), (T, H), (T, T)\}$

In general, $A×B≠B×A$.
If $m$ is the number of elements in $A$ and $n$ is the number of elements in $B$, then the number of elements in $A×B$ or $B×A$ is $mn$.
If $R$ is the set of real numbers, then the Cartesian Product of $R$ on $R$ i.e. $R×R$ or $R^2$ is the set $\{(x,y):x∈R\;\text{and}\;y∈R\}$. This Cartesian Product is represented by the entire Cartesian Coordinate Plane.

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