Which of the following functions from A to B are one-one and onto?

Solution:

(i) $f_{1}=\{(1,3),(2,5),(3,7)\} ; A=\{1,2,3\}, B=\{3,5,7\}$

Injectivity:

$f_{1}(1)=3$

$f_{1}(2)=5$

$f_{1}(3)=7$

$\Rightarrow$ Every element of $A$ has different images in $B$.

So, $f_{1}$ is one-one.

Surjectivity:

Co-domain of $f_{1}=\{3,5,7\}$

Range of $f_{1}=$ set of images $=\{3,5,7\}$

$\Rightarrow$ Co-domain $=$ range

So, $f_{1}$ is onto.

(ii) f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}

Injectivity:

$f_{2}(2)=a$

$f_{2}(3)=b$

$f_{2}(4)=c$

$\Rightarrow$ Every element of $A$ has different images in $B$.

So, $f_{2}$ is one-one.

$\Rightarrow$ Every element of $A$ has different images in $B$.

So, $f_{2}$ is one-one.

Surjectivity:

Co-domain of $f_{2}=\{a, b, c\}$

Range of $f_{2}=$ set of images $=\{a, b, c\}$

$\Rightarrow$ Co-domain $=$ range

So, $f_{2}$ is onto.

(iii) $f_{3}=\{(a, x),(b, x),(c, z),(d, z)\} ; A=\{a, b, c, d,\}, B=\{x, y, z$,

Injectivity:

$f_{3}(a)=x$

$f_{3}(b)=x$

$f_{3}(c)=z$

$f_{3}(d)=z$

$\Rightarrow a$ and $b$ have the same image $x$. (Also $c$ and $d$ have the same image $z$ )

So, $f_{3}$ is not one-one.

Surjectivity:

Co-domain of $f_{1}=\{x, y, z\}$

Range of $f_{1}=$ set of images $=\{x, z\}$

So, the co-domain is not same as the range

So, $f_{3}$ is not onto.