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# Question-2

To find the set of relations in the directed graph and then the inverse relations, we can follow these steps:

1. **Identify the Set of Relations:**
Each directed edge (arrow) in the graph represents a relation from one node to another. We will list all these relations as ordered pairs \((a, b)\) where there is a directed edge from node \(a\) to node \(b\).

2. **Inverse Relations:**
The inverse of a relation \((a, b)\) is \((b, a)\). We will invert each relation identified in step 1.

3. **Redraw the Directed Graph:**
Using the inverse relations, we will redraw the directed graph.

### Step 1: Identify the Set of Relations
From the directed graph, we can identify the following relations (edges):
- (1, 2)
- (1, 4)
- (2, 3)
- (3, 4)
- (4, 1)
- (4, 5)
- (4, 7)
- (5, 6)
- (6, 5)

### Step 2: Inverse Relations
The inverse relations are:
- (2, 1)
- (4, 1)
- (3, 2)
- (4, 3)
- (1, 4)
- (5, 4)
- (7, 4)
- (6, 5)
- (5, 6)

### Step 3: Redraw the Directed Graph
To redraw the directed graph with the inverse relations, we simply reverse the direction of each arrow based on the inverse relations identified.

I will now create a textual description for redrawing the directed graph with the inverse relations and generate the image.

### Textual Description for the Inverted Graph
- Node 2 has a directed edge to Node 1.
- Node 4 has directed edges to Nodes 1 and 3.
- Node 3 has a directed edge to Node 2.
- Node 1 has a directed edge to Node 4.
- Node 5 has directed edges to Nodes 4 and 6.
- Node 6 has a directed edge to Node 5.
- Node 7 has a directed edge to Node 4.

I will use this information to generate the redrawn directed graph with the inverse relations.

# Question-3

Here's the redrawn directed graph based on the inverse relations from the original graph. Let me know if you need any further assistance!

To analyze the given graph, we need to evaluate it based on several criteria: number of vertices, number of edges, degree sequence, parallel edges, total degree, loops, and then construct a new graph with specified degrees.

### Step-by-Step Analysis

1. **Number of Vertices:**
The vertices are \( v1, v2, v3, v4, v6, v7 \).
Number of vertices \( |V| = 6 \).

2. **Number of Edges:**
Count all the directed edges.
Number of edges \( |E| = 9 \).

3. **Degree Sequence:**
- Degree of \( v1 \): Out-degree = 3 (edges to \( v2, v3, v4 \)), In-degree = 2 (edges from \( v4, v6 \)) -> Degree = 5
- Degree of \( v2 \): Out-degree = 1 (edge to \( v1 \)), In-degree = 2 (edges from \( v1, v7 \)) -> Degree = 3
- Degree of \( v3 \): Out-degree = 1 (edge to \( v2 \)), In-degree = 1 (edge from \( v1 \)) -> Degree = 2
- Degree of \( v4 \): Out-degree = 2 (edges to \( v1, v6 \)), In-degree = 1 (edge from \( v7 \)) -> Degree = 3
- Degree of \( v6 \): Out-degree = 1 (edge to \( v1 \)), In-degree = 2 (edges from \( v4 \)) -> Degree = 3
- Degree of \( v7 \): Out-degree = 2 (edges to \( v2, v4 \)), In-degree = 1 (loop to itself) -> Degree = 3

Degree sequence: \( 5, 3, 3, 3, 3, 2 \).

4. **Parallel Edges:**
There are no parallel edges (multiple edges between the same pair of vertices).

5. **Total Degree:**
Total degree is the sum of the degrees of all vertices: \( 5 + 3 + 3 + 3 + 3 + 2 = 19 \).

6. **Loops:**
There is one loop on vertex \( v7 \).

### Construct a Graph with 5 Vertices of Degrees 1, 1, 2, 2, 4

To construct a graph with the specified degrees, we can consider a simple example:

1. **Vertices:** \( A, B, C, D, E \)
2. **Degrees:**
- Degree 1: \( A \)
- Degree 1: \( B \)
- Degree 2: \( C, D \)
- Degree 4: \( E \)

Constructing the graph:
- \( E \) should be connected to all other vertices to satisfy degree 4.
- \( C \) and \( D \) should be connected to \( E \) and one more vertex each to satisfy degree 2.
- \( A \) and \( B \) should be connected to \( E \) to satisfy degree 1.

Graph: \( A - E, B - E, C - E, D - E, C - D \)

I will now draw both the given graph and the constructed graph as draw.io files.

# Question 4

To determine if the given graphs (G1, G2, G3, G4, and G5) are isomorphic, we need to check if they are structurally the same. Two graphs are isomorphic if there is a one-to-one correspondence between their vertex sets that preserves adjacency.

### Steps to Determine Isomorphism:
1. **Compare the number of vertices and edges.**
2. **Compare the degree sequences.**
3. **Check for a one-to-one correspondence that preserves adjacency.**

Let's analyze each graph:

1. **G1:**
- Vertices: 6
- Edges: 9
- Degree sequence: 3, 3, 3, 3, 3, 3

2. **G2:**
- Vertices: 6
- Edges: 9
- Degree sequence: 3, 3, 3, 3, 3, 3

3. **G3:**
- Vertices: 6
- Edges: 9
- Degree sequence: 3, 3, 3, 3, 3, 3

4. **G4:**
- Vertices: 6
- Edges: 9
- Degree sequence: 3, 3, 3, 3, 3, 3

5. **G5:**
- Vertices: 6
- Edges: 9
- Degree sequence: 3, 3, 3, 3, 3, 3

Since all graphs have the same number of vertices, edges, and degree sequences, we proceed to check their adjacency:

1. **G1:**
- Edges: (0-1), (0-2), (0-3), (1-2), (1-4), (2-5), (3-4), (3-5), (4-5)

2. **G2:**
- Edges: (0-1), (0-2), (0-3), (1-2), (1-4), (2-5), (3-4), (3-5), (4-5)

3. **G3:**
- Edges: (0-1), (0-2), (0-3), (1-2), (1-4), (2-5), (3-4), (3-5), (4-5)

4. **G4:**
- Edges: (0-1), (0-2), (0-3), (1-2), (1-4), (2-5), (3-4), (3-5), (4-5)

5. **G5:**
- Edges: (0-1), (0-2), (0-3), (1-2), (1-4), (2-5), (3-4), (3-5), (4-5)

### Conclusion:
Based on the adjacency and degree sequence, all graphs are isomorphic as they have the same structure. The visual difference in drawing doesn't affect the structural isomorphism. Thus, G1, G2, G3, G4, and G5 are isomorphic graphs.

If you have any specific criteria or additional steps you'd like me to consider, please let me know!

# Answers of Suggestions-

Let's address each part of the question step by step.

### 1. Definitions

**Graph:**
A graph \(G\) consists of a set of vertices \(V(G)\) and a set of edges \(E(G)\). Each edge is a pair of vertices, representing a connection between them. Graphs can be directed or undirected.

**Subgraph:**
A subgraph of a graph \(G\) is a graph \(H\) such that the vertices of \(H\) are a subset of the vertices of \(G\) and the edges of \(H\) are a subset of the edges of \(G\).

**Edge:**
An edge is a connection between two vertices in a graph. In a directed graph, edges have a direction, going from one vertex to another. In an undirected graph, edges do not have a direction.

**Vertex:**
A vertex (or node) is a fundamental part of a graph, representing a point where edges meet. Vertices are usually denoted by letters or numbers.

**Degree:**
The degree of a vertex in a graph is the number of edges incident to (connected to) the vertex. In a directed graph, we distinguish between in-degree (number of incoming edges) and out-degree (number of outgoing edges).

**Loop:**
A loop is an edge that connects a vertex to itself. It contributes twice to the degree of the vertex (once as an incoming edge and once as an outgoing edge in a directed graph).

**Isolated Vertex:**
An isolated vertex is a vertex that has no edges connected to it. Its degree is zero.

### 2. Prove the Sum of the Degree Theorem

The sum of the degree theorem states that in any graph, the sum of the degrees of all the vertices is equal to twice the number of edges.

**Proof:**

Let's consider an undirected graph \(G = (V, E)\) with \(V\) being the set of vertices and \(E\) being the set of edges.

1. **Sum of Degrees:**
The degree of a vertex is the number of edges incident to it. If we sum the degrees of all the vertices in the graph, each edge will be counted twice (once for each endpoint of the edge).

2. **Mathematical Representation:**
Let \(d(v)\) be the degree of vertex \(v\). The sum of the degrees of all vertices is:


3. **Counting Edges:**
Each edge \(e \in E\) connects two vertices. Therefore, each edge contributes 2 to the total sum of degrees (once for each endpoint).

4. **Relation to Edges:**
Since each edge is counted twice in the sum of degrees, we have:


where \(|E|\) is the number of edges in the graph.

### Example:
Consider a graph with 4 vertices \(A, B, C,\) and \(D\) and 3 edges \((A-B, B-C, C-D)\).

- Degrees: \(d(A) = 1\), \(d(B) = 2\), \(d(C) = 2\), \(d(D) = 1\)
- Sum of degrees: \(1 + 2 + 2 + 1 = 6\)
- Number of edges: 3
- According to the theorem: Sum of degrees \( = 2 \times \text{Number of edges} = 2 \times 3 = 6\)

Thus, the sum of the degrees theorem is validated.

This completes the definitions and proof. If you need any further explanation or examples, feel free to ask!

## What is Homo Morphism-
The graphs are homomorphic if the number of vertices and edges are the same, and the degree

## Question 9

To classify the function $f: \mathbb{R} \to \mathbb{R}^+$ defined by $f(x) = 3x - 1$, we need to examine its properties and behavior.

### Steps for Classification:
1. **Domain and Range:**
- The function $f(x) = 3x - 1$ is defined for all real numbers$x$.
- The range of $f$ is the set of positive real numbers $\mathbb{R}^+$, which implies $f(x) > 0$.

2. **Behavior and Properties:**
- **Linearity:** The function $f(x) = 3x - 1$ is a linear function since it can be written in the form $f(x) = ax + b$ with $a = 3$ and $b = -1$.
- **Monotonicity:** To determine if the function is increasing or decreasing, we look at the coefficient of $x$:
- Since $ 3 > 0 $, the function is strictly increasing.
- **Zeros:** The function $f(x) = 3x - 1$ has a zero when $3x - 1 = 0$:
$\[
3x - 1 = 0 \implies x = \frac{1}{3}
\]$
- **Range:** For $f(x) to be positive, $3x - 1 > 0$:
$\[
3x - 1 > 0 \implies x > \frac{1}{3}
\]$
This means f(x) maps $x > \frac{1}{3}$ to positive real numbers.

3. **Classify Based on Range:**
- The function f(x) has a range of positive real numbers only if $x > \frac{1}{3}$.
- Thus, $f: \mathbb{R} \to \mathbb{R}^+$ cannot be valid for all $x \in \mathbb{R}$, but it is valid for $x \in \left( \frac{1}{3}, \infty \right)$.

### Classification and Justification:
- **Linear Function:** The function is a linear function because it has the form $f(x) = ax + b$.
- **Strictly Increasing:** The function is strictly increasing because the coefficient of x is positive.
- **Valid Range:** The function is only valid for $x > \frac{1}{3}$ to ensure the output is positive, satisfying $f(x) \in \mathbb{R}^+$.

### Final Classification:
The function $f(x) = 3x - 1$ is a linear, strictly increasing function valid for $x \in \left( \frac{1}{3}, \infty \right)$. It maps this domain to the positive real numbers $\mathbb{R}^+$.

### Another Answer:

To classify the function \( f: \mathbb{R} \to \mathbb{R}^+ \) defined by \( f(x) = 3x - 1 \) in terms of properties such as symmetry, transitivity, reflexivity, anti-symmetry, and equivalence, we need to interpret these properties in the context of a function. These properties are usually used to describe relations rather than functions. However, we can discuss whether the function \( f \) induces any such relation on \(\mathbb{R}\) based on its properties.

### Symmetric:
A relation \( R \) on a set \( A \) is symmetric if \( (a, b) \in R \) implies \( (b, a) \in R \).

For a function \( f \), this would mean that if \( f(a) = b \), then \( f(b) = a \). However, for the function \( f(x) = 3x - 1 \), this is not generally true. For example, if \( f(1) = 2 \), it is not the case that \( f(2) = 1 \). Therefore, the function \( f \) is **not symmetric**.

### Transitive:
A relation \( R \) on a set \( A \) is transitive if \( (a, b) \in R \) and \( (b, c) \in R \) imply \( (a, c) \in R \).

For functions, this property does not typically apply in a straightforward way. Functions are mappings and not relations between pairs of elements in the same set, so this property is not applicable here.

### Reflexive:
A relation \( R \) on a set \( A \) is reflexive if \( (a, a) \in R \) for all \( a \in A \).

For a function, this would mean that \( f(a) = a \) for all \( a \). Clearly, \( f(x) = 3x - 1 \) does not satisfy \( f(x) = x \) for all \( x \). Therefore, the function \( f \) is **not reflexive**.

### Anti-Symmetric:
A relation \( R \) on a set \( A \) is anti-symmetric if \( (a, b) \in R \) and \( (b, a) \in R \) imply \( a = b \).

For a function \( f \), this means that if \( f(a) = b \) and \( f(b) = a \), then \( a = b \). Given \( f(x) = 3x - 1 \), if \( f(a) = b \), then \( b = 3a - 1 \), and if \( f(b) = a \), then \( a = 3b - 1 \). Solving these equations together leads to a contradiction unless \( a = b \). Therefore, the function \( f \) is **anti-symmetric**.

### Equivalence:
A relation \( R \) on a set \( A \) is an equivalence relation if it is reflexive, symmetric, and transitive.

Since the function \( f(x) = 3x - 1 \) is not reflexive or symmetric, it cannot define an equivalence relation. Therefore, \( f \) does **not define an equivalence relation**.

### Summary:
- The function \( f(x) = 3x - 1 \) is not symmetric.
- The concept of transitivity does not straightforwardly apply to functions.
- The function \( f(x) = 3x - 1 \) is not reflexive.
- The function \( f(x) = 3x - 1 \) is anti-symmetric.
- The function \( f(x) = 3x - 1 \) does not define an equivalence relation.

These classifications demonstrate that while some properties apply more naturally to relations, we can still derive meaningful interpretations for functions in certain contexts.