Question

Let \[X=\left\{ a,b,c,d \right\}\] 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
1. \[\left\{ \left( a,a \right),\left( b,c \right),\left( c,d \right),\left( d,b \right) \right\}\]
2. \[\left\{ \left( a,a \right),\left( b,d \right),\left( c,c \right),\left( d,b \right) \right\}\]
3. \[\left\{ \left( a,a \right),\left( b,d \right),\left( c,b \right),\left( d,c \right) \right\}\]
A. Only (1) is true.
B. (1) and (2) is true.
C. (1), (2) and (3) are true.
D. Only (3) is true.

Answer 1

Hint: In this problem, we have to find the correct options by mapping in one-one function. We are given a set X, which can be mapped into one-one function as we know that one-one maps distinct elements of its domain to distinct elements of its codomain. We can then check the options which satisfy the given condition and find the required answer.

Complete step by step answer:
We are given that \[X=\left\{ a,b,c,d \right\}\].
We are also given \[f:X\to X\]. We can now map it in one-one function as we know that one-one maps distinct elements of its domain to distinct elements of its codomain.

We are given that \[f\left( a \right)=a,f\left( b \right)\ne b,f\left( d \right)\ne d\].
We can now take 1) and check whether the given condition is satisfied.
1. \[\left\{ \left( a,a \right),\left( b,c \right),\left( c,d \right),\left( d,b \right) \right\}\]
2. \[\left\{ \left( a,a \right),\left( b,d \right),\left( c,c \right),\left( d,b \right) \right\}\]
3. \[\left\{ \left( a,a \right),\left( b,d \right),\left( c,b \right),\left( d,c \right) \right\}\]

Here we can see that the first set, second set and the third set satisfy the given condition \[f\left( a \right)=a,f\left( b \right)\ne b,f\left( d \right)\ne d\].

So, the correct answer is “Option C”.

Note: We should always remember that one-one maps distinct elements of its domain to distinct elements of its codomain. We should also know that a one-one function is also called an injective function, where the function’s codomain is the image of at most one element of its domain.