1. Introduction to Boolean Logic
Boolean algebra uses binary variables that have two states: 0 (False) and 1 (True). It uses logical operations: AND, OR, and NOT.
Named after mathematician George Boole.
2. Basic Logic Gates
| Gate | Symbol | Expression | Truth Table |
|---|---|---|---|
| AND | . | Y = A · B | 0·0=0, 0·1=0, 1·0=0, 1·1=1 |
| OR | + | Y = A + B | 0+0=0, 0+1=1, 1+0=1, 1+1=1 |
| NOT | ' or ¬ | Y = A' | 0'=1, 1'=0 |
3. Universal and Derived Gates
| Gate | Expression | Truth Table |
|---|---|---|
| NAND | Y = (A·B)' | 0·0=1, 0·1=1, 1·0=1, 1·1=0 |
| NOR | Y = (A+B)' | 0+0=1, 0+1=0, 1+0=0, 1+1=0 |
| XOR | Y = A ⊕ B | 0⊕0=0, 0⊕1=1, 1⊕0=1, 1⊕1=0 |
| XNOR | Y = (A⊕B)' | 0⊕0=1, 0⊕1=0, 1⊕0=0, 1⊕1=1 |
NAND and NOR are called universal gates because any logic gate can be constructed using only NAND or only NOR gates.
4. Boolean Algebra Laws
| Law | AND Form | OR Form |
|---|---|---|
| Identity Law | A · 1 = A | A + 0 = A |
| Null Law | A · 0 = 0 | A + 1 = 1 |
| Idempotent Law | A · A = A | A + A = A |
| Complement Law | A · A' = 0 | A + A' = 1 |
| Double Negation | (A')' = A | |
| Commutative Law | A · B = B · A | A + B = B + A |
| Associative Law | (A·B)·C = A·(B·C) | (A+B)+C = A+(B+C) |
| Distributive Law | A·(B+C)=A·B+A·C | A+B·C=(A+B)·(A+C) |
5. De Morgan's Laws
De Morgan's laws provide a way to simplify Boolean expressions:
1. (A · B)' = A' + B'
(NAND is equivalent to OR with inverted inputs)
2. (A + B)' = A' · B'
(NOR is equivalent to AND with inverted inputs)6. Boolean Expression and Simplification
Example: Simplify F = A·B + A·B' + A'·B
= A·B + A·B' + A'·B
= A·(B + B') + A'·B
= A·(1) + A'·B
= A + A'·B
= (A + A')·(A + B) [Distributive law]
= 1·(A + B)
= A + BNote: A + A'·B = A + B is a standard reduction.
7. Revision Questions and Answers
Very Short Answer Questions
1. What are the three basic logic gates?
AND, OR, NOT.
2. Which gates are called universal gates?
NAND and NOR.
3. What is the output of 1 XOR 1?
0.
4. State De Morgan's first law.
(A·B)' = A' + B'.
5. What does A + A' equal?
1.
Short Answer Questions
1. Prove De Morgan's second law using a truth table.
(A+B)' = A'·B'. Truth table: For A=0,B=0: (0+0)'=1, 0'·0'=1·1=1. A=0,B=1: (0+1)'=0, 0'·1'=1·0=0. A=1,B=0: (1+0)'=0, 1'·0'=0·1=0. A=1,B=1: (1+1)'=0, 1'·1'=0·0=0. Both match.
2. Simplify F = A·(A + B).
= A·A + A·B = A + A·B = A·(1+B) = A·1 = A.
Long Answer Questions
1. Write the truth table, Boolean expression, and logic circuit symbol for AND, OR, NOT, NAND, NOR, XOR, and XNOR gates.
Refer to sections above for truth tables. For CBSE, focus on: AND (•), OR (+), NOT ('), NAND (↓), NOR (↑), XOR (⊕). Circuit symbols: AND has flat left, curved right; OR has curved left and right; NOT has triangle with bubble at output; NAND = AND with bubble; NOR = OR with bubble; XOR = OR with extra curve on left.
2. Simplify F = A'·B·C + A·B'·C + A·B·C' + A·B·C using Boolean algebra.
= C·(A'·B + A·B') + A·B·(C' + C)
= C·(A ⊕ B) + A·B·1
= C·(A⊕B) + A·B