Question

Let $X=\{a,b,c,d\}$ then one – one mapping $f:X\to X$ such that $f\left(a\right)=a,f\left(b\right)\ne b,f\left(d\right)\ne d$ are given by

$\left(\mathrm{i}\right)\left\{\left(a,a\right),\left(b,c\right),\left(c,d\right),\left(d,b\right)\right\}$

$\left(\mathrm{ii}\right)\left\{\left(a,a\right),\left(b,d\right),\left(c,c\right),\left(d,b\right)\right\}$

$\left(\mathrm{iii}\right)\left\{\left(a,a\right),\left(b,d\right),\left(c,b\right),\left(d,c\right)\right\}$

• A. only (i) is true
• B. (i) and (ii) is true
• C. (i), (ii) and (iii) are true
• D. only (iii) is true

Answer 1

The correct option is C

(i), (ii) and (iii) are true

Explanation for the correct answer:

Finding all the possible relations of the given function:

The possibilities of the function for the set $X=\{a,b,c,d\}$ by one-one mapping $f:X\to X$ such that $f\left(a\right)=a,f\left(b\right)\ne b,f\left(d\right)\ne d$ are given by,

Using the concept of one-to-one function

$\left(\mathrm{i}\right)\left\{\left(a,a\right),\left(b,c\right),\left(c,d\right),\left(d,b\right)\right\}$

$\left(\mathrm{ii}\right)\left\{\left(a,a\right),\left(b,d\right),\left(c,c\right),\left(d,b\right)\right\}$

$\left(\mathrm{iii}\right)\left\{\left(a,a\right),\left(b,d\right),\left(c,b\right),\left(d,c\right)\right\}$

Therefore, the correct answer is option (C).