Let R be the reduction from 3-SAT to VERTEX-COVER as discussed in class. **Graph Explanation:**The graph shown is a bipartite graph consisting of two sets of nodes, connected by edges. The nodes are labeled as \( P, \neg P, Q, \neg Q, S, \neg S, T, \neg T \). The edges between the nodes indicate the relationships that need to be considered to construct the 3-CNF (Conjunctive Normal Form) formula.### Nodes:- **Positive Nodes:** \( P, Q, S, T \)- **Negative Nodes:** \( \neg P, \neg Q, \neg S, \neg T \)### Edges:The graph connects each variable and its negation:- \( P \) is connected to \( \neg P \)- \( Q \) is connected to \( \neg Q \)- \( S \) is connected to \( \neg S \)- \( T \) is connected to \( \neg T \)Additionally, other connections exist indicating clause construction for the formula:- Edges between positive and negative nodes suggest combinations in clauses.**Question:**(1) Show the 3-CNF formula for which R builds the following graph.**Answer Choices:**- \( \circ \ (P \land Q \land T) \lor (P \land Q \land S) \lor (T \land Q \land S) \)- \( \circ \ (P \lor Q \lor T) \land (P \lor Q \lor S) \land (T \lor Q \lor S) \)- \( \circ \ (\neg P \land Q \land T) \lor (\neg P \land Q \land S) \lor (T \land \neg Q \land S) \)- \( \circ \ (\neg P \lor Q \lor T) \land (\neg P \lor Q \lor S) \land (T \lor \neg Q \lor S) \)

In 3 CNF SAT we have at least 3 clauses and in the clauses we will have the almost 3 literals.3 CNF formula for which R builds given graph is:(P V Q V T) (P V Q V S) (T V Q V S)CONCLUSION:Correct option: 4

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(1) Show the 3-CNF formula for which R builds the following graph. P -P T S -P 0 (P^Q ^T) V (P^Q ^S) V (T ^Q ^S) O (P V QVT) A (PVQVS) A (TV Q V S) O (PAQAT) V (P AQ AS) V (TA QAS) O (PV QVT) A (PV QV S) A (TV QV S) -S S T -T S

(1) Show the 3-CNF formula for which R builds the following graph. P -P T S -P 0 (P^Q ^T) V (P^Q ^S) V (T ^Q ^S) O (P V QVT) A (PVQVS) A (TV Q V S) O (PAQAT) V (P AQ AS) V (TA QAS) O (PV QVT) A (PV QV S) A (TV QV S) -S S T -T S

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Let R be the reduction from 3-SAT to VERTEX-COVER as discussed in class.

**Graph Explanation:**

The graph shown is a bipartite graph consisting of two sets of nodes, connected by edges. The nodes are labeled as \( P, \neg P, Q, \neg Q, S, \neg S, T, \neg T \). The edges between the nodes indicate the relationships that need to be considered to construct the 3-CNF (Conjunctive Normal Form) formula.

### Nodes:

- **Positive Nodes:** \( P, Q, S, T \)
- **Negative Nodes:** \( \neg P, \neg Q, \neg S, \neg T \)

### Edges:

The graph connects each variable and its negation:

- \( P \) is connected to \( \neg P \)
- \( Q \) is connected to \( \neg Q \)
- \( S \) is connected to \( \neg S \)
- \( T \) is connected to \( \neg T \)

Additionally, other connections exist indicating clause construction for the formula:

- Edges between positive and negative nodes suggest combinations in clauses.

**Question:**

(1) Show the 3-CNF formula for which R builds the following graph.

**Answer Choices:**

- \( \circ \ (P \land Q \land T) \lor (P \land Q \land S) \lor (T \land Q \land S) \)
- \( \circ \ (P \lor Q \lor T) \land (P \lor Q \lor S) \land (T \lor Q \lor S) \)
- \( \circ \ (\neg P \land Q \land T) \lor (\neg P \land Q \land S) \lor (T \land \neg Q \land S) \)
- \( \circ \ (\neg P \lor Q \lor T) \land (\neg P \lor Q \lor S) \land (T \lor \neg Q \lor S) \)

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