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Topic 5 of 25
05
Propositions and Truth Tables
Propositions and Truth Tables
Propositions and Truth Tables
1. Propositions
A proposition is a declarative sentence that is either true (T) or false (F)—not both.
Examples:
- "7 is a prime number." → True
- "5 + 3 = 9." → False
Propositions are represented by symbols like , and are the building blocks of propositional logic.
2. Logical Connectives
Logical connectives are used to form compound propositions from simple ones. The main ones are:
| Symbol | Operation | Meaning |
|---|---|---|
| Negation | Not | |
| Conjunction | and | |
| Disjunction | or | |
| Implication | If , then | |
| Biconditional | if and only if | |
| Exclusive OR | or , but not both |
3. Truth Tables
A truth table shows all possible truth values of a compound proposition based on the truth values of its components.
a) Negation
| p | ¬p |
|---|---|
| T | F |
| F | T |
b) Conjunction
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
c) Disjunction
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
d) Implication
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
e) Biconditional
| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
f) Exclusive OR
| p | q | p ⊕ q |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
4. Truth Table for a Compound Proposition
Example: Construct the truth table for
| p | q | ¬p | p ∨ q | (p ∨ q) ∧ ¬p |
|---|---|---|---|---|
| T | T | F | T | F |
| T | F | F | T | F |
| F | T | T | T | T |
| F | F | T | F | F |
5. Number of Rows in a Truth Table
For propositional variables, the truth table has rows.
- For 1 variable: rows
- For 2 variables: rows
- For 3 variables: rows
- And so on...
6. Use of Truth Tables
- Verify equivalences (e.g., )
- Determine tautologies, contradictions, and contingencies
- Analyze logical arguments
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