We are given the set $S = \{1, 2, 3\}$ and the set $P_3 = \{ f : S \to S \}$, which represents the set of all functions from $S$ to $S$. We are tasked with finding the elements of $P_3$, particularly with respect to function composition.

Step 1: Understanding $P_3$

Each function $f : S \to S$ is a mapping of the elements of $S = \{1, 2, 3\}$ to itself. Since $S$ has three elements, and each element of $S$ can be mapped to any of the three elements, there are $3 \times 3 \times 3 = 27$ possible functions from $S$ to $S$.

Step 2: Listing All Functions

Each element in $P_3$ is a function $f : S \to S$. To find all elements of $P_3$, we can represent each function as a set of ordered pairs where each input (from $S$) is mapped to an output (also from $S$).

A few examples of such functions are:

1. $f_1: 1 \mapsto 1, 2 \mapsto 2, 3 \mapsto 3$ (the identity function)
2. $f_2: 1 \mapsto 1, 2 \mapsto 2, 3 \mapsto 1$
3. $f_3: 1 \mapsto 2, 2 \mapsto 3, 3 \mapsto 1$

Since each function defines a unique mapping of $S$ to $S$, and there are 27 such mappings, $P_3$ contains 27 distinct functions.

Step 3: Composition of Functions

The composition of two functions $f$ and $g$, denoted $f \circ g$, is defined as:

$(f \circ g)(x) = f(g(x))$

For instance, if $f(1) = 2$ and $g(2) = 3$, then $(f \circ g)(1) = f(g(1)) = f(2) = 3$. Therefore, to compute function compositions, you would systematically apply one function after another.

Step 4: Conclusion

The set $P_3$ consists of 27 functions, each being a different mapping from $S = \{1, 2, 3\}$ to itself. The elements of $P_3$ are these 27 functions, and function composition is defined as the standard composition of two such functions.

Would you like more details on function composition or examples of specific compositions between functions?

5 Related Questions:

1. How do we determine if a function is invertible from $S \to S$?
2. How can we represent each function in $P_3$ using a matrix?
3. What are the properties of the identity function under composition?
4. How do we compute the composition of multiple functions in $P_3$?
5. Can we group the functions in $P_3$ by specific properties (e.g., constant, identity)?

Tip:

When computing compositions of functions, always evaluate the inner function first and then apply the outer function.