Question

Write the total number of one-one functions from the set \[A=\text{ }\left\{ 1,2,3,4 \right\}\]to the set \[B=\text{ }\left\{ a,b,c \right\}\].

Answer 1

Hint: In this case, we are required to find the number of valid functions from the set A to B. Therefore, we should understand the definition and properties of a function and particularly an one-one function from one set to another. Thereafter, we can check which functions can be defined from A to B given in this question.

Complete step-by-step answer:
We need to check the total number of one-one functions from A to B. Therefore, we need to first understand the definition of a function which is stated as follows
A function can be defined as a binary relation between two states which associates each element of the first set to exactly one element of the second set…………………………………. (1.1)
A one-one function from a set A to a set B associates each element of A to exactly one element of B and no element of B is associated to more than one element of A………………………………. (1.2)


From, definition (1.2), we see that as each element of A has to be associated to an element of B and each element of B can be associated with only element of A, the number of elements of B must be greater than or equal to the number of elements of A……………………………………. (1.3)
In this case the set A is given by $A=\left\{ 1,2,3,4 \right\}$which contains 4 elements and the set B is given by $B=\left\{ a,b,c \right\}$ which contains 3 elements. Thus, the number of elements of B is less than that of A……………. (1.4)
Now, if there exists a one-one function from A to B, then it must satisfy equation (1.3) which contradicts equation (1.4). Thus, there is no such function possible and the answer to this question should be zero.

Note: It is important to consider the property of one-one because there can be normal functions from A to B which are not one-one, for example, the function which associates 1 and 2 to a, 3 to b and 4 to c is a valid function but is not one-one.