Mathematical Logic is one of the most important and scoring topics in the UGC NET Computer Science (CS) exam. It forms the foundation for reasoning, programming, artificial intelligence, and formal verification. In this guide, we’ll cover Propositional Logic and Predicate Logic in a structured and exam-oriented way.
📌 What is Mathematical Logic?
Mathematical Logic deals with formal systems of reasoning. It allows us to represent statements and analyze their truth values using symbols and rules.
It is broadly divided into:
- Propositional Logic
- Predicate Logic
🔹 Propositional Logic
✅ Definition
Propositional Logic (also called sentential logic) deals with propositions—statements that are either true (T) or false (F).
Example:
- “2 + 2 = 4” → True
- “India is in Europe” → False
🔑 Logical Connectives
| Symbol | Name | Example |
|---|---|---|
| ¬P | Negation | Not P |
| P ∧ Q | AND | P and Q |
| P ∨ Q | OR | P or Q |
| P → Q | Implication | If P then Q |
| P ↔ Q | Biconditional | P iff Q |
📊 Truth Tables
Truth tables are used to evaluate logical expressions.
Example: AND (P ∧ Q)
| P | Q | P ∧ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
⚡ Important Concepts
1. Tautology
Always true
Example: P ∨ ¬P
2. Contradiction
Always false
Example: P ∧ ¬P
3. Contingency
Sometimes true, sometimes false
🔄 Logical Equivalences
These are very important for UGC NET:
- De Morgan’s Laws
- ¬(P ∧ Q) = ¬P ∨ ¬Q
- ¬(P ∨ Q) = ¬P ∧ ¬Q
- Implication Law
- P → Q = ¬P ∨ Q
📌 Normal Forms
1. Conjunctive Normal Form (CNF)
AND of ORs
2. Disjunctive Normal Form (DNF)
OR of ANDs
🔹 Predicate Logic (First Order Logic)
✅ Definition
Predicate Logic extends propositional logic by including:
- Variables
- Quantifiers
- Predicates
🔑 Components
1. Predicate
Represents a property or relation
Example: P(x): “x is even”
2. Quantifiers
| Symbol | Meaning |
|---|---|
| ∀x | For all x |
| ∃x | There exists x |
Example:
- ∀x P(x) → All elements satisfy P
- ∃x P(x) → At least one element satisfies P
🔄 Negation Rules
- ¬(∀x P(x)) = ∃x ¬P(x)
- ¬(∃x P(x)) = ∀x ¬P(x)
⚡ Free and Bound Variables
- Bound Variable: Inside quantifier
- Free Variable: Not bound
📌 Important Topics for UGC NET
You must focus on:
- Logical equivalence
- Valid arguments
- Rules of inference
- Predicate transformation
- Skolemization
- Prenex Normal Form
🔹 Rules of Inference
Used to derive conclusions:
- Modus Ponens
- If P → Q and P is true → Q is true
- Modus Tollens
- If P → Q and Q is false → P is false
🎯 UGC NET Exam Tips
- Practice previous year questions (PYQs)
- Focus on truth tables & equivalence laws
- Revise quantifiers and negations carefully
- Solve MCQs daily
📚 Conclusion
Mathematical Logic is a high-weightage and scoring topic in UGC NET Computer Science. A strong grasp of Propositional Logic and Predicate Logic can significantly improve your score.
Mastering concepts like truth tables, logical equivalences, quantifiers, and inference rules will give you a clear advantage in the exam.
