Sets, Relations, and Functions form the backbone of Discrete Mathematics, which is a core part of the UGC NET Computer Science syllabus (Unit 1: Discrete Structures and Optimization). These topics are highly scoring and frequently asked in the exam.
This guide will help you understand all concepts in a clear, structured, and exam-oriented way.
🔹 Sets
✅ What is a Set?
A set is a well-defined collection of distinct objects.
Example:
- A = {1, 2, 3, 4}
- B = {a, e, i, o, u}
🔑 Types of Sets
- Finite Set – Limited elements
- Infinite Set – Unlimited elements
- Empty Set (∅) – No elements
- Universal Set (U) – All possible elements
- Subset (⊆) – A ⊆ B
🔄 Set Operations
| Operation | Symbol | Meaning |
|---|---|---|
| Union | A ∪ B | Elements in A or B |
| Intersection | A ∩ B | Common elements |
| Difference | A − B | Elements in A not in B |
| Complement | A’ | Elements not in A |
⚡ Important Laws
- Commutative: A ∪ B = B ∪ A
- Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Distributive:
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
📊 Power Set
The power set is the set of all subsets.
- If a set has n elements → Power set has 2ⁿ elements
🔹 Relations
✅ Definition
A relation R from set A to B is a subset of A × B (Cartesian product).
🔑 Types of Relations
1. Reflexive
(a, a) ∈ R for all a ∈ A
2. Symmetric
If (a, b) ∈ R → (b, a) ∈ R
3. Transitive
If (a, b) ∈ R and (b, c) ∈ R → (a, c) ∈ R
⚡ Equivalence Relation
A relation that is:
- Reflexive
- Symmetric
- Transitive
👉 It partitions a set into equivalence classes
🔄 Representation of Relations
- Roster form
- Matrix form
- Directed graph (Digraph)
🔹 Functions
✅ Definition
A function is a special type of relation where each element of domain has exactly one image in codomain.
🔑 Types of Functions
1. One-One (Injective)
Different inputs → Different outputs
2. Onto (Surjective)
Every element in codomain is mapped
3. Bijective
Both one-one and onto
🔄 Special Functions
- Identity Function: f(x) = x
- Constant Function: f(x) = c
🔁 Composition of Functions
- (f ∘ g)(x) = f(g(x))
🔄 Inverse Function
- Exists only if function is bijective
📌 Important Concepts for UGC NET
Focus on these high-weightage areas:
- Set identities and laws
- Power set problems
- Types of relations (especially equivalence)
- Closure properties
- Function types (injective, surjective, bijective)
- Composition and inverse
🎯 Exam Tips
- Practice Venn diagram-based questions
- Solve previous year questions (PYQs)
- Focus on relation properties (very common in exam)
- Practice identifying function types quickly
📚 Conclusion
Sets, Relations, and Functions are foundational topics in UGC NET Computer Science. With proper practice and conceptual clarity, you can easily score full marks from this section.
Mastering these concepts will also help in advanced topics like graph theory, databases, and automata theory.
