🌟 2.1 LOGIC GATES
Logic gates are electronic circuits that perform logical operations on binary inputs (0 or 1).
1️⃣ Basic Gates
AND Gate
| A | B | Output = A • B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
OR Gate
| A | B | A + B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
NOT Gate
| A | A' |
|---|---|
| 0 | 1 |
| 1 | 0 |
2️⃣ Universal Gates (NAND, NOR)
Any circuit can be built using only NAND or only NOR.
NAND Gate
Output = (A • B)’
Truth Table → Only 1 when both inputs are NOT 1.
NOR Gate
Output = (A + B)’
Truth Table → Only 1 when both inputs are 0.
3️⃣ Exclusive Gates
XOR Gate
Output = 1 when inputs are different.
Equation: A ⊕ B = A’B + AB’
XNOR Gate
Output = 1 when inputs are same.
🌟 2.2 BOOLEAN ALGEBRA
Important Laws
1. Identity Laws
A + 0 = A
A • 1 = A
2. Null Laws
A + 1 = 1
A • 0 = 0
3. Idempotent Laws
A + A = A
A • A = A
4. Complement Laws
A + A’ = 1
A • A’ = 0
5. De Morgan’s Laws
(AB)’ = A’ + B’
(A + B)’ = A’B’
🌟 2.3 Minterms & Maxterms
Minterm (SOP Form)
Each minterm = product term (AND)
Example for 2 variables:
m0 = A’B’
m1 = A’B
m2 = AB’
m3 = AB
Maxterm (POS Form)
Each maxterm = sum term (OR)
M0 = A + B
M1 = A + B’ etc.
🌟 2.4 Simplification Using Karnaugh Map (K-MAP)
Example:
Simplify F(A, B, C) = Σ(1, 3, 5, 7)
K-MAP:
Fill 1’s in cells 1,3,5,7.
Groups formed:
-
Group of four (1,3,5,7)
Final simplified function:
👉 F = B
K-map makes long Boolean expressions very small.
🌟 2.5 NUMBER CODES
✔ 1. Weighted Codes
Each digit has a weight.
Examples:
-
8421 (BCD)
-
2421
-
5211
✔ 2. Non-Weighted Codes
Weights do NOT apply.
Examples:
Gray Code
Only one bit changes between consecutive numbers (used in error-free encoding).
Binary to Gray:
G1 = B1
G2 = B1 ⊕ B2
G3 = B2 ⊕ B3
Example:
Binary 101 → Gray = 111
Excess-3 Code
Add 3 to each BCD digit.
Example:
BCD 0101 (5)
Excess-3 = 0101 + 0011 = 1000
Code Conversions Covered
✔ Binary ↔ Gray
✔ Binary ↔ BCD
✔ BCD ↔ Excess-3
✔ Hex ↔ Binary
✔ Decimal ↔ Binary
No comments:
Post a Comment