Question

Let f : {x,y,z} $\to$ {a,b,c} be a one-one function. It is known that only one of the following statment is true, and only one such function exists :

find the function f (as ordered pair).(i) f(x) $\ne$ b
(i) f(y) = b

(ii) f(z) $\ne$ a

• A. {(x,b), (y,a), (z,c)}
• B. {(x,a), (y,b), (z,c)}
• C. {(x,b), (y,c), (z,a)}
• D. {(x,c), (y,a), (z,b)}

Answer 1

The correct option is B {(x,a), (y,b), (z,c)}

When (i) is true, then $f\left(x\right)\ne b,f\left(y\right)\ne b,f\left(z\right)=a,b,c$

$\Rightarrow$ Two ordered pair function is possible $f\left(x\right)=a,f\left(y\right)=c,f\left(z\right)=b$ or $f\left(x\right)=c,f\left(y\right)=a,f\left(z\right)=b$

But given $f$ is one-one and only one such function is possible. Hence (i) can't be true.

When (ii) is true, then $f\left(y\right)=b,f\left(z\right)=c,f\left(x\right)=a\Rightarrow f\left(x\right)=a,f\left(y\right)=b,f\left(z\right)=c$

Clearly if (ii) is true then it is satisfying every condition.

Hence ordered pair of $f$ is $\left\{\right(x,a),(y,b),(z,c\left)\right\}$