Question

Set A has 3 elements and set B has 4 elements. The number of injections that can be defined from A into B is:

& A.144 \\\ & B.12 \\\ & C.24 \\\ & D.64 \\\ \end{aligned}$$

Answer 1

In this question, we are given a number of elements of two sets A and B and we have to find the number of injections that can be defined from A to B. This means we have to find the number of one-one functions from A into B. For this, we will first understand one one function and how to use them. After that, we will find a number of ways function can be defined.

Complete step by step answer:
Here, we are given two sets A and B having 3 and 4 elements respectively. We have to find the number of injections that can be defined from A to B. Let us first understand the meaning of injection. Injections are functions that map distinct elements of its domain to distinct elements of its codomain or we can say that every element of the functions codomain is the image of at most one element of its domain. In this sum, we can thus say that, to define injective function from set A to B, we can map the first element of set A to any of the 4 elements of set B. Then the second element cannot be mapped to same element of set A, hence there are three choices in set B for the second element of set A. Similarly, there are two choices in set B for the third elements of set A. From all this, we conclude that, total number of injection from set A to set B are $4\times 3\times 2=24$.
Hence, total injections from set A into set B are 24.

So, the correct answer is “Option C”.

Note: Students should not get confused with injections, surjective and bijection. Injection refers to one to one function, surjective refers to onto function and bijection refers to one to one and onto function. Students should keep in mind that there is a difference between injections from A to B and from B to A. For B to A, there exists no injection, since B has more numbers of elements than A.