Find the number of all one-one functions from set $A=\{1{,}2{,}3\}$ to itself.

Step 1: Understand the Definition of a One-One Function

A one-one function, also known as an injective function, maps distinct elements of the domain to distinct elements of the codomain. In simpler terms, no two different elements in the domain can have the same image in the codomain.

Step 2: Determine the Domain and Codomain

The problem states that the function maps from set $A$ to itself. Therefore, both the domain and the codomain are the set $A = \{1{,}2{,}3\}$. The number of elements in set $A$ is $n=3$.

Step 3: Map the First Element

For the first element in the domain, which is 1, there are 3 possible choices in the codomain (1, 2, or 3) to which it can be mapped.

Step 4: Map the Second Element

Since the function must be one-one, the second element in the domain, 2, cannot be mapped to the same element as $f(1)$. Therefore, there are only 2 remaining choices in the codomain for $f(2)$.

Step 5: Map the Third Element

Similarly, for the third element in the domain, 3, it cannot be mapped to the same elements as $f(1)$ or $f(2)$. This leaves only 1 remaining choice in the codomain for $f(3)$.

Step 6: Calculate the Total Number of One-One Functions

To find the total number of one-one functions, we multiply the number of choices for each element. This is equivalent to calculating the factorial of the number of elements in the set, which is $3!$.

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