๐ง Truth Tables
Truth tables are used to represent the output of logical expressions for all possible input combinations. Each row corresponds to a unique input state, and the columns show the resulting output.
๐ค Notation Legend
| Symbol | Meaning |
|---|---|
ยฌA | NOT A (negation) |
A โง B | A AND B (conjunction) |
A โจ B | A OR B (disjunction) |
A โ B | A XOR B (exclusive OR) |
A โ B | A IMPLIES B |
A โ B | A IFF B (biconditional) |
โ Basic Truth Tables
1๏ธโฃ NOT (Negation)
| A | ยฌA |
|---|---|
| 0 | 1 |
| 1 | 0 |
2๏ธโฃ AND (Conjunction)
| A | B | A โง B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
3๏ธโฃ OR (Disjunction)
| A | B | A โจ B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
4๏ธโฃ XOR (Exclusive OR)
| A | B | A โ B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
๐ Compound Expressions
Example: (A โง B) โจ ยฌC
| A | B | C | A โง B | ยฌC | (A โง B) โจ ยฌC |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 0 | 1 |
๐งฉ Tips for Building Truth Tables
- List all input combinations using binary counting.
- Compute intermediate columns for sub-expressions.
- Use consistent notation and spacing for clarity.
- Highlight final output column for emphasis.
Check: ๐ Baseโb Recursive Listing Method
๐งฎ Expandable Framework
To add more operators or multi-bit logic, duplicate the table structure and extend with additional columns. For example:
- NAND:
ยฌ(A โง B) - NOR:
ยฌ(A โจ B) - XNOR:
ยฌ(A โ B) - Majority:
(A โง B) โจ (A โง C) โจ (B โง C)
๐ง Set-Theoretic Analogy (Optional)
A โง Bโ Intersection: $ A \cap B $A โจ Bโ Union: $ A \cup B $ยฌAโ Complement: $ A^c $A โ Bโ Symmetric Difference: $ A \triangle B $
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