Functions
Introduction :
The famous Mathematician "LEJEUNE DIRICHLET" (1805 - 1859) defined a function as follows. Let x and y are any two variables, if they are so related that whenever a value is assigned to x by some rule or correspondence a value to y, then we say y is a function of x i.e., y = f(x)
Ordered pair :
Let A and B be two sets, then \(
\left\{ {\left( {a,b} \right)/a \in A,\,\,b \in B} \right\}
\) is called the cartesian product of A and B is denoted by A x B (A cross B)
EX : if A ={1,2,3}, B = { x, y }
A x B = {(1,x), (1,y), (2,x), (2,y), (3,x), (3,y)}
B x A = {(x,1),(x,2), (x,3), (y,1), (y,2),(y,3)}
Note :\(
A \times B \ne \,B \times A
\)
Relation :
If A and B are non-empty sets, then any subset of A xB is called a relation from A to B. In particular, in any relation from A to A is called binary relation on A
Ex : A = {1,2,3}, B = {\(
\alpha ,\beta
\)} \(
A \times B = \{ (1,\alpha ),(1,\beta ),(2,\alpha ),(2,\beta ),(3,\alpha ),(3,\beta )\}
\) then
Let \(
f = \{ (1,\alpha ),(3,\alpha )\}
\) is a relation from A to B.
\(
g = \{ (1,\beta ),(2,\alpha ),(3,\alpha ),(3,\beta )
\) is also another relation from A to B
Note : If n(A) =m, n(B)=n, then the number of relation from A to B = 2mxn from the above example
The number of relation = 23x2 = 26 = 64