DIGITAL LOGIC & DISCRETE MATHEMATICS (FULL COVERAGE)

January 04, 2026

1️⃣ NUMBER SYSTEMS (VERY HIGH WEIGHTAGE)

🔸 What is a Number System?

A way to represent numbers using a base (radix).

System	Base	Digits Used
Decimal	10	0–9
Binary	2	0,1
Octal	8	0–7
Hexadecimal	16	0–9, A–F

🔸 Binary ↔ Decimal Conversion (MOST ASKED)

Binary → Decimal

Multiply each bit with powers of 2 (right to left)

Example:
1011₂
= 1×2³ + 0×2² + 1×2¹ + 1×2⁰
= 11₁₀

Decimal → Binary

Divide by 2 repeatedly, write remainders bottom → top

Example:
25₁₀ = 11001₂

🔸 Octal & Hex Conversion (Direct MCQs)

Octal → Binary

Each digit = 3 bits

Octal	Binary
0	000
7	111

Hex → Binary

Each digit = 4 bits

Hex	Binary
A	1010
F	1111

📌 Shortcut MCQ rule:
Hex ↔ Binary = direct replacement

🔸 Important MCQs

Largest digit in octal → 7
Base of hexadecimal → 16
Binary of A → 1010

2️⃣ LOGIC GATES (GUARANTEED QUESTIONS)

🔸 What is a Logic Gate?

An electronic circuit that performs logical operations.

🔸 Basic Gates & Rules

Gate	Rule
AND	Output = 1 if all inputs are 1
OR	Output = 1 if any input is 1
NOT	Output is inverse

🔸 Truth Tables (MUST MEMORIZE)

AND Gate

A	B	Y
0	0	0
0	1	0
1	0	0
1	1	1

🔸 Universal Gates ⭐⭐⭐

Gate	Why Universal
NAND	Can implement all gates
NOR	Can implement all gates

📌 MCQ:
Which gate is universal? → NAND / NOR

🔸 XOR Gate (CONFUSION MAKER)

Output = 1 only when inputs are different

A	B	XOR
0	0	0
0	1	1
1	0	1
1	1	0

3️⃣ BOOLEAN ALGEBRA (MOST SCORING)

🔸 Boolean Variables

Only two values: 0 or 1

🔸 Important Laws (EXAM FAVORITE)

Law	Expression
Identity	A + 0 = A
Null	A + 1 = 1
Idempotent	A + A = A
Complement	A + Ā = 1
Commutative	A + B = B + A
Associative	(A + B) + C
Distributive	A(B + C)

🔥 De Morgan’s Theorem (VERY VERY IMPORTANT)

1️⃣ (A + B)̄ = Ā · B̄
2️⃣ (A · B)̄ = Ā + B̄

📌 MCQ:
Which theorem converts OR to AND? → De Morgan

🔸 Simplification Example

(A + ĀB) = (A + B)

📌 MCQs often ask final simplified form

4️⃣ COMBINATIONAL vs SEQUENTIAL CIRCUITS

Feature	Combinational	Sequential
Output depends on	Present input	Input + past
Memory	❌ No	✅ Yes
Examples	Adder, MUX	Flip-flop, Counter

📌 Golden Rule:
Memory present → Sequential

🔸 Flip-Flops (BASIC ONLY)

Flip-Flop	Key Point
SR	Invalid when S = R = 1
JK	No invalid state
D	Stores data
T	Toggle

📌 Most repeated MCQ:
Which flip-flop has no invalid state? → JK

5️⃣ SETS (DIRECT QUESTIONS)

🔸 Definition

A collection of well-defined objects.

🔸 Types of Sets

Type	Meaning
Empty	No elements
Finite	Limited
Infinite	Unlimited
Universal	All elements

🔸 Operations

Union (∪) → All elements
Intersection (∩) → Common elements
Difference (−)

📌 MCQ:
A ∩ B contains? → Common elements

6️⃣ RELATIONS (CONFUSION BUT EASY)

A relation is a subset of Cartesian product.

🔸 Properties of Relation

Property	Meaning
Reflexive	aRa
Symmetric	aRb ⇒ bRa
Transitive	aRb & bRc ⇒ aRc

📌 MCQ:
If aRb and bRa → Symmetric

7️⃣ FUNCTIONS (GUARANTEED THEORY MCQs)

🔸 What is a Function?

Each input has exactly one output.

🔸 Types of Functions

Type	Meaning
One-One (Injective)	Unique output
Onto (Surjective)	All outputs covered
Many-One	Allowed
Into	Not all outputs used

📌 MCQ:

Injective = One-One
Surjective = Onto

🎯 MOST EXPECTED EXAM MCQs (FINAL CHECK)

1️⃣ Universal gate? → NAND
2️⃣ Output differs → XOR
3️⃣ Memory circuit → Sequential
4️⃣ OR → AND law → De Morgan
5️⃣ No invalid FF → JK
6️⃣ aRb ⇒ bRa → Symmetric

🧠 ONE-PAGE MEMORY RULES (EXAM GOLD)

Binary → base 2
Universal → NAND / NOR
Memory → Sequential
Flip-flop no invalid → JK
OR ↔ AND → De Morgan
Common set → Intersection