Question

The number of one-one functions that can be defined from set {a, b, c, d} to the set {1, 2, 3, 4} is?

Answer 1

Hint: Use the cardinality of two sets to find the number of one-one functions between them.
$\text{Number of one-one functions = }{}^{n}{{P}_{m}}\text{ if n}\ge \text{m}$…..(1)
Number of one-one functions = 0 if n < m…..(2)

Complete step-by-step answer:
The total number of elements in the set is called the cardinality of the set.
Let us assume given sets as A and B, that is,
A = {a, b, c, d} and B = {1, 2, 3, 4}
The cardinality of a set is denoted by “|set|”
Here cardinality of A = |A| = 4.
Cardinality of B = |B| = 4.
If there are two non-empty sets with cardinality m and n, then the number of one-one functions between them is given by:
$\text{Number of one-one functions = }{}^{n}{{P}_{m}}\text{ if n}\ge \text{m}$ …..(1)
Number of one-one functions = 0 if n < m…..(2)
By the above formula, in our case the value of m is 4 and the value of n is 4.
We can see that m = n.
So we need to use equation (1):
$\text{Number of one-one functions = }{}^{n}{{P}_{m}}\text{ if n}\ge \text{m}$.
So, the number of one-one functions = ${}^{4}{{P}_{4}}$
By using the formula,
${}^{n}{{P}_{m}}=\dfrac{n!}{\left( n-m \right)!}$
By substituting factorial of 0 as 1,we get:
$\text{The number of one-one functions }={}^{4}{{P}_{4}}=\dfrac{4!}{(4-4)!}=4!$ [$\because$ 0!=1]

The number of one-one functions = (4)(3)(2)(1) = 24.
$\therefore$The total number of one-one functions from {a, b, c, d} to {1, 2, 3, 4} is 24.

Note: Here the values of m, n are same but in case they are different then the direction of checking matters. If m > n, then the number of one-one from first set to the second becomes 0. So take care of the direction of checking.