Property 5
Function
Let A and B be two non-empty sets. A relation which matches each element of A to one and only one element of B is called a function from A to B.
It is denoted as \(f:A\to B\)
- Set A is called the domain
- Set B is called the range
- \(f(A)=\{1,3,4\}\) is called the image set.
Which of the following relation is a function?
1.
\(A=\{1,2,3\}\)
\(B=\{a,b,c\}\)
\(\beta:A\to B\)
\(\mathrm{I.}\ \beta=\{(1,a),(2,b),(3,c)\}\)
\(\mathrm{II.}\ \beta=\{(1,a),(2,a),(3,a)\}\)
\(\mathrm{III.}\ \beta=\{(1,a),(2,b),(3,c),(1,b)\}\)
\(\mathrm{IV.}\ \beta=\{(1,a),(3,b)\}\)
2.
\(\beta:\mathbb{R}\to \mathbb{R}\)
\(\mathrm{I.}\ \beta=\{(x,y)\mid x=2\}\)
\(\mathrm{II.}\ \beta=\{(x,y)\mid x^2+y^2=4\}\)
\(\mathrm{III.}\ \beta=\{(x,y)\mid y=x^2+2\}\)
\(\mathrm{IV.}\ \beta=\{(x,y)\mid|x|+|y|=2\}\)
Property 6
Let \(f:A\to B\) be a function.
- \(\text{For every }x_1,x_2\in A\)
\(f(x_1)=f(x_2)\Rightarrow x_1=x_2\)
If proposition is true then it is said that f is a one to one function.
- If \(f(A)=B\), then it is said that f is an onto function
Example
\(\text{f is a one to one and onto function}\)
\(\begin{gathered}\text{f is one to one but not onto}\\\text{function}\end{gathered}\)
\(\begin{gathered}\text{f is onto but not one to one}\\\text{function}\end{gathered}\)
Analyse the characteristics one to one or onto function which are given below.
1.
\(A=\{1,2,3\}\)
\(B=\{a,b,c\}\)
\(f=\{(1,a),(2,b),(3,c)\}\)
2.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=2x+1\)
3.
\(f:\mathbb{Z}\to \mathbb{Z}\)
\(f(x)=3x-1\)
4.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=x^2\)