Let X be a set with four elements. Represent the identity function 1X of Example 2.22 
(1X : X → X is defined by 1X(x) = x.)
 with a directed graph in two different ways.
(a) with four vertices, each representing an element in both the domain and the codomain

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.

Let X be a set with four elements. Represent the identity function 1X of Example 2.22 (1X : X → X is defined by 1X(x) = x.) with a directed graph in two different ways. (a) with four vertices, each representing an element in both the domain and the codomain

Understand how to represent the identity function for a set with four elements using a directed graph. Learn about graph theory concepts, including vertices and edges, in the context of the identity function, suitable for high school students in Grades 9-12.

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