Ordered Pairs and the Cartesian Product

Before a relation or a function can be defined precisely, we need a way of pairing up elements from two sets so that the order in which they are written matters. An ordered pair $(a, b)$ consists of a first element $a$ and a second element $b$, written in that fixed order; two ordered pairs $(a,b)$ and $(c,d)$ are equal only when $a=c$ and $b=d$. This is the key difference from an ordinary set $\{a,b\}$, where $\{a,b\}$ and $\{b,a\}$ are the same set — but $(a,b)$ and $(b,a)$ are, in general, two different ordered pairs.

Given two non-empty sets $A$ and $B$, the Cartesian product $A \times B$ is the set of all ordered pairs whose first element comes from $A$ and whose second element comes from $B$:

If $A$ has $m$ elements and $B$ has $n$ elements, then $A \times B$ has exactly $m \times n$ ordered pairs — every element of $A$ is paired once with every element of $B$. For example, if $A=\{1,2\}$ and $B=\{x,y,z\}$, then

which has $2 \times 3 = 6$ ordered pairs, exactly as the counting rule predicts. In general $A \times B \neq B \times A$ unless $A=B$, because reversing the order of every pair produces a genuinely different set of pairs.

The Odisha CHSE Std-11 Business Mathematics & Statistics syllabus builds the whole chapter on Functions on top of this one idea — a relation, and later a function, is nothing more than a carefully chosen subset of a Cartesian product $A \times B$.