To represent the identity function $1_X : X \to X$ with a directed graph where $X$ is a set of four elements, let's follow the steps outlined in part (a) to illustrate the identity function graphically.

Part (a): Directed Graph with Four Vertices

1. Vertices: The set $X$ consists of four elements, say $X = \{x_1, x_2, x_3, x_4\}$. The graph will have four vertices, each corresponding to an element of $X$.

2. Edges: Since $1_X(x) = x$, each element in the set maps to itself. Hence, there will be a loop from each vertex to itself, representing the fact that each element is mapped to itself under the identity function.

Thus, the graph has:

• Four vertices, $x_1, x_2, x_3, x_4$,
• Each vertex has a self-loop, indicating that $1_X(x_i) = x_i$ for all $i = 1, 2, 3, 4$.

Visual Structure of the Graph:

• Vertices: $x_1, x_2, x_3, x_4$,
• Edges: A directed loop (edge) from each vertex to itself:
  • From $x_1$ to $x_1$,
  • From $x_2$ to $x_2$,
  • From $x_3$ to $x_3$,
  • From $x_4$ to $x_4$.

This graph visually demonstrates that each element in $X$ is mapped to itself in the identity function.

If you'd like, I can generate a visual diagram of this graph. Would you like to see a graphical representation? Also, feel free to ask for further details or explanations!

Here are 5 questions to extend this concept:

1. How would the graph change if $X$ had more than four elements?
2. What would the directed graph look like if it represented a non-identity function from $X$ to $X$?
3. Can the identity function on a set ever produce a directed graph without loops?
4. How is the concept of identity function used in real-world applications of graph theory?
5. How does the graph of an identity function differ from that of a permutation function on the same set?

Tip: The identity function always preserves the structure of the set it's defined on—each element maps to itself, which is why the loops in the graph exist.