Question

Given X=\left\\{ 1,2,3,4 \right\\}, find all one-one onto mapping $f:X\to X$ such that $f\left( 1 \right)=1$, $f\left( 2 \right)\ne 2$
and $f\left( 4 \right)\ne 4$. (This question has multiple correct options).
(a) \left\\{ \left( 1,1 \right),\left( 2,3 \right),\left( 3,4 \right),\left( 4,2 \right) \right\\}
(b) \left\\{ \left( 1,1 \right),\left( 2,4 \right),\left( 3,3 \right),\left( 4,2 \right) \right\\}
(c) \left\\{ \left( 1,1 \right),\left( 2,4 \right),\left( 3,2 \right),\left( 4,3 \right) \right\\}
(d) \left\\{ \left( 1,4 \right),\left( 2,1 \right),\left( 3,2 \right),\left( 4,3 \right) \right\\}

Answer 1

We are given a set and a function on the set with certain conditions. This question has multiple correct options. Hence, we will check whether each and every option satisfies the conditions of the function. We will also verify that every mapping given in the options is one-one and onto.

Complete step by step answer:
Given set is X=\left\\{ 1,2,3,4 \right\\}, and the function is $f:X\to X$ such that $f\left( 1 \right)=1$, $f\left( 2 \right)\ne 2$ and $f\left( 4 \right)\ne 4$. As there are multiple correct options for this question, we will check every option one by one. Let us start with option (a). The given mapping in this choice is \left\\{ \left( 1,1 \right),\left( 2,3 \right),\left( 3,4 \right),\left( 4,2 \right) \right\\}. There are 4 elements in the set $X$. We can see that every element in the domain is mapped to every element in the range. Hence, the mapping is one-one and onto. Next, we will check the conditions on the function. We can see that $f\left( 1 \right)=1$ and $f\left( 2 \right)=3\ne 2$, also $f\left( 4 \right)=2\ne 4$. So, the mapping in option (a) satisfies the given conditions.
Now, we will look at option (b). The given mapping is \left\\{ \left( 1,1 \right),\left( 2,4 \right),\left( 3,3 \right),\left( 4,2 \right) \right\\}. All 4 elements of the domain are mapped to all 4 elements of the range distinctly. Hence, the mapping is one-one and onto. We can see that this mapping satisfies the conditions $f\left( 1 \right)=1$, $f\left( 2 \right)\ne 2$ and $f\left( 4 \right)\ne 4$. Next is option (c). The given mapping is \left\\{ \left( 1,1 \right),\left( 2,4 \right),\left( 3,2 \right),\left( 4,3 \right) \right\\}. Since every element in the domain is mapped to every element in the range, the mapping is one-one and onto. We can see that this mapping satisfies the given conditions $f\left( 1 \right)=1$, $f\left( 2 \right)\ne 2$ and $f\left( 4 \right)\ne 4$.
Now, we will check option (d). The given mapping is ) \left\\{ \left( 1,4 \right),\left( 2,1 \right),\left( 3,2 \right),\left( 4,3 \right) \right\\}. Every element in the domain is mapped to every element in the range. Hence, the mapping is one-one and onto. We see that $f\left( 1 \right)=4$. According to the given conditions, $f\left( 1 \right)=1$. This implies that option (d) does not satisfy the given conditions.

So, the correct answer is “Option A, B and C”.

Note: It is essential that we understand the one-one and onto mappings. Questions where there are multiple correct options can be tricky since we need to identify all the correct options. In these questions, the conditions on the functions were fairly straightforward and hence easy to check for all options. If the conditions become more complex, we need to be careful while checking the options.