Given two nonempty sets \(A\) and \(B\). The Cartesian product \(A \times B\) is
the set of all ordered pairs of elements from \(A\) and \(B\), i.e.,
\(A \times B = \left\{ {\left( {a,b} \right):a \in A,b \in B} \right\}\)
If either \(A\) or \(B\) is the null set, then \(A \times B\) will also be
empty set, i.e., \(A \times B\)
Questtion 1
If \(P = \left\{ {1,2} \right\}\) and \(Q = \left\{ {5,4,2} \right\}\), find \(P \times Q\) and \(Q \times P\).
Solution
\(P = \left\{ {1,2} \right\}\) and \(Q = \left\{ {5,4,2} \right\}\)
We know that the Cartesian product \(P \times Q\) of two nonempty sets \(P\) and \(Q\) is defined as
\(P \times Q = \left\{ {\left( {p,q} \right):p \in P,q \in Q} \right\}\)
\(\therefore P \times Q = \left\{ {\left( {1,5} \right),\left( {1,4} \right),\left( {1,2} \right),\left( {2,5} \right),\left( {2,4} \right),\left( {2,2} \right)} \right\}\)
\(Q \times P = \left\{ {\left( {5,1} \right),\left( {5,2} \right),\left( {4,1} \right),\left( {4,2} \right),\left( {2,1} \right),\left( {2,2} \right)} \right\}\)
Question 2
If \(A = \left\{ {  1,1} \right\}\), find \(A \times A \times A\)
Solution
It is known that for any nonempty set \(A\), \(A \times A \times A\) is defined as
\(A \times A \times A = \left\{ {\left( {a,b,c} \right):a,b,c \in A} \right\}\)
It is given that \(A = \left\{ {  1,1} \right\}\)
\(\therefore A \times A \times A\left\{ {\left( {  1,  1,  1} \right),\left( {  1,  1,1} \right),\left( {  1,1,  1} \right),\left( {  1,1,1} \right),\left( {1,  1,  1} \right),\left( {1,  1,1} \right),\left( {1,1,  1} \right),\left( {1,1,1} \right),} \right\}\)
A relation \(R\) from a nonempty set \(A\) to a nonempty set \(B\) is a subset of the cartesian product \(A \times B\).
It "maps" elements of one set to another set. The subset is derived by describing a relationship
between the first element and the second element of the ordered pair \(\left( {A \times B} \right)\).
Domain: The set of all first elements of the ordered pairs in a relation \(R\) from a set
\(A\) to a set \(B\) is called the domain of the relation \(R\).
Range: the set of all the ending points is called the range
A relation can be expressed in Set builder or Roaster form
In a Roster forms, all the elements in the set is listed.
Example
Set of \(vovel = \left\{ {a,e,i,o,u} \right\}\)
The total number of relations that can be defined from a set \(A\) to a set \(B\) is the number of possible subsets of \(A \cdot B\). If \(n\left( A \right) = p\) and \(n\left( B \right) = q\), then \(n\left( {A \cdot B} \right) = pq\) and the total number of relations is \({2^{pq}}\)
Example:
Let \(P = \left\{ {1,2,3,.....,18} \right\}\) define a relation \(R\) from \(P\) to \(P\) by \(R = \left\{ {\left( {x,y} \right):2x  y = 0,where{\rm{ }}x,y \in P} \right\}\) Write down its domain, codomain and range.
Solution:
The relation \(R\) from \(P\) to \(P\) is given as
R = {(x,y):2xy=0, where x, y ∈ P}
i.e., R = {(x, y): 2x = y, where x, y ∈ P}
\(\therefore R{\rm{ }} = {\rm{ }}\left\{ {\left( {1,{\rm{ }}2} \right),{\rm{ }}\left( {2,{\rm{ }}4} \right),{\rm{ }}\left( {3,{\rm{ }}6} \right),{\rm{ }}\left( {4,{\rm{ }}8} \right),{\rm{ }}\left( {5,{\rm{ }}10} \right),{\rm{ }}\left( {6,{\rm{ }}12} \right),{\rm{ }}\left( {7,{\rm{ }}14} \right),{\rm{ }}\left( {8,{\rm{ }}16} \right),\left( {9,18} \right)} \right\}\)
The domain of \(R\) is the set of all first elements of the ordered pairs in the relation.
\(\therefore Domain{\rm{ }}of{\rm{ }}R{\rm{ }} = {\rm{ }}\left\{ {1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}4,5,6,7,8,9} \right\}\)
The whole set \(P\) is the codomain of the relation \(R\).
Therefore codomain of \(R{\rm{ }} = {\rm{ }}P{\rm{ }} = {\rm{ }}\left\{ {1,{\rm{ }}2,{\rm{ }}3,{\rm{ }} \ldots ,{\rm{ }}18} \right\}\)
The range of \(R\) is the set of all second elements of the ordered pairs in the relation.
Therefore range of \(R = \left\{ {2,{\rm{ }}4,{\rm{ }}6,{\rm{ }}8,10,12,14,16,18} \right\}\)
Example 1:
Which of the following relations are functions? Give reasons. If it is a function, determine
its domain and range.
Let us take some useful polynomial and shapes obtained on the Cartesian plane
S.No.  \(y = p\left( x \right)\)  Graph obtained  Name of the graph  Name of the function 
1.  \(y = mx + c\) where m and c can be any
values \(\left( {m \ne 0} \right)\) Example \(y = 2x + 3\) 
Graphs of these functions are straight lines. \(m\) is the slope and \(b\) is the \(y\) intercept. If \(m\) is positive then the line rises to the right and if \(m\) is negative then the line falls to the right 
Linear function. Typical use for linear functions is converting from one quantity or set of units to another. 

2.  \(y = a{x^2} + bx + c\) where, \({b^2}  4ac{\rm{ }} > {\rm{ }}0\) , \(a \ne 0\) and \(a > {\rm{ }}0\) example \(y = {x^2}  7x + 12\) 
Parabola It intersect the x axis at two points Example (3,0) and (4,0) 
Quadratic function  
3.  \(y = a{x^3} + b{x^2} + cx + d\) where,\(a \ne 0\) 
It can be of any shape 
It will cut the xaxis at the most 3 times  Cubic Function 
4.  \({a_n}{x^n} + {a_{n  1}}{x^{n  1}} + {a_{n  2}}{x^{n  2}} + \ldots \ldots \ldots \ldots + ax + {a_0}\) where \({a_n} \ne 0\) 
It can be of any shape 
It will cut the xaxis at the most n times  Polynomial function 
5.  \(y = \frac{{f(x)}}{{g(x)}}\) where \(g(x) \ne 0\) example \(y = \frac{1}{x}\) 
It can be any shape 
An asymptote is a line that the curve approaches but does not cross.There are vertical and horizontal asymptote 
Rational function 
6.  \(y = \left x \right\) i.e., \(y =  x\) for \(x < 0\) \(y = x\) for \(x \ge 0\) 
Modulus function  
7.  \(y = aln\left( x \right) + b\) where \(x\) is in the natural logarithm and \(a\) and \(b\) are constants They are only defined for positive 
For small \(x\) they are negative and for large \(x\) they are positive \(x\)  Logarithmic functions  
8.  \(y = \left[ x \right]\) \(\left[ x \right]{\rm{ }}  > \)the value of the greatest integer, less than or equal to x 
Greatest integer function 
Real Value Function: A function which has all real number or subset of the real number as it domain
Real Valued Function: A function which has all real number or subset of the real number as it range
For functions \(f:{\rm{ }}X  > {\bf{R}}\) and \(g:{\rm{ }}X  > {\bf{R}}\), we have