1.0 Definitions

1.1 Cartesian Product of Sets

An ordered pair consists of two objects or elements in a given fixed order.

If $A$ and $B$ are any two sets, then by an ordered pair of elements, it means a pair $(a,b)$ in that order, where $a \in A$, $b \in B$.

Equality of Ordered Pairs:

Two ordered pairs $({a_1},{b_1})$ and $({a_2},{b_2})$ are equal if and only if ${a_1} = {a_2}$ and ${b_1} = {b_2}$.

i.e. $$({a_1},{b_1}) = ({a_2},{b_2}) \Leftrightarrow \;{a_1} = {a_2}\;\;and\;\;{b_1} = {b_2}$$

Example 1. Find $x$, $y$ , $z$ in each case $$\begin{equation} \begin{aligned} i.\;\;(x + 1, - 1) = (4,y + 6) \\ ii.\;\;({x^2},27) = (9,{y^3}) \\ iii.\;({y \over 4} + 1,x - {3 \over 4}) = (3,1) \\ iv.\;(x + y,7) = (2,2x - y) \\\end{aligned} \end{equation} $$

$i.\;\;(x + 1, - 1) = (4,y + 6)$

${\rm{By}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{equality}}\;{\rm{of}}\;{\rm{ordered}}\;{\rm{pairs}}$

$$\begin{equation} \begin{aligned} \Rightarrow x + 1 = 4 \Rightarrow x = 3 \\ \Rightarrow y + 6 = - 1 \Rightarrow y = - 7 \\ Thus\;x = 3\;and\;y = - 7 \\\end{aligned} \end{equation} $$

$ii.\;\;({x^2},27) = (9,{y^3})$

${\rm{By}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{equality}}\;{\rm{of}}\;{\rm{ordered}}\;{\rm{pairs}}$

$$\begin{equation} \begin{aligned} \Rightarrow {x^2} = 9\; \Rightarrow \;x = \pm 3 \\ \Rightarrow {y^3} = 27 \Rightarrow y = 3 \\ Thus\;x = \pm 3\;\;and\;y = 3 \\\end{aligned} \end{equation} $$

$iii.\;({y \over 4} + 1,x - {3 \over 4}) = (3,1)$

${\rm{By}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{equality}}\;{\rm{of}}\;{\rm{ordered}}\;{\rm{pairs}}$

$$\begin{equation} \begin{aligned} \Rightarrow {y \over 4} + 1 = 3 \Rightarrow y + 4 = 12 \Rightarrow y = 8 \\ \Rightarrow x - {3 \over 4} = 1 \Rightarrow 4x - 3 = 4 \Rightarrow x = {7 \over 4} \\ Thus,\;x = {7 \over 4}\;{\mkern 1mu} and\;y = 8 \\\end{aligned} \end{equation} $$

$iv.\;(x + y,7) = (2,2x - y)$

${\rm{By}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{equality}}\;{\rm{of}}\;{\rm{ordered}}\;{\rm{pairs}}$ $$\begin{equation} \begin{aligned} \Rightarrow x + y = 2 \Rightarrow y = 2 - x \\ \Rightarrow 2x - y = 7 \Rightarrow 2x - (2 - x) = 7 \\ \Rightarrow 3x - 2 = 7 \Rightarrow 3x = 9 \Rightarrow x = 3 \\ \Rightarrow y = 2 - 3 = - 1 \\ Thus,\;x = 3\;and\;y = - 1 \\\end{aligned} \end{equation} $$

Example 2. Write all possible ordered pairs $(a,b)$ where $a \in A$, where $A = \{ 1,3,5\} $ and $b \in B$, where $B = \{ 2,4,6\} $

Solution: The ordered pairs are $(1,2),(1,4),(1,6),(3,2),(3,4),(3,6),(5,2),(5,4),(5,6)$.