Basic and Derived Logical Operations: Connectives and Truth Tables
Explore basic logical operations (AND, OR, NOT, implication, equivalence) and how they combine propositions to form compound statements. This guide includes truth tables to illustrate the truth values of these operations and shows how derived operations can be expressed using basic connectives.
Basic and Derived Logical Operations
Basic Logical Connectives
Logical connectives combine propositions (statements that are either true or false) to create more complex statements. Here are the five basic connectives:
1. Negation (¬)
The negation of a statement p (¬p, also written ~p, p', or -p) is true when p is false and false when p is true.
| p | ¬p |
|---|---|
| True | False |
| False | True |
2. Conjunction (∧)
The conjunction of statements p and q (p ∧ q, read as "p and q") is true only when both p and q are true.
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
3. Disjunction (∨)
The disjunction of statements p and q (p ∨ q, read as "p or q") is true if at least one of p or q is true.
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
4. Implication (→)
The implication "if p, then q" (p → q) is false only when p is true and q is false.
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
5. Biconditional (↔)
The biconditional "p if and only if q" (p ↔ q) is true when p and q have the same truth value (both true or both false).
| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Derived Logical Connectives
1. NAND (↑)
NAND (p ↑ q) is the negation of AND. It's false only when both p and q are true.
| p | q | p ↑ q |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
2. NOR (↓)
NOR (p ↓ q) is the negation of OR. It's true only when both p and q are false.
| p | q | p ↓ q |
|---|---|---|
| T | T | F |
| T | F | F |
| F | T | F |
| F | F | T |
3. XOR (⊕)
XOR (p ⊕ q) is true when exactly one of p or q is true.
| p | q | p ⊕ q |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
Examples: Proving Logical Equivalences
Example 1: XOR Equivalence
(The proof showing the equivalence of X ⊕ Y and (X ∧ ¬Y) ∨ (¬X ∧ Y) using truth tables is provided in the original text and should be included here.)
Example 2: Equivalence with NAND and NOR
(The proof showing the equivalence of (p ⊕ q) ∨ (p ↓ q) and p ↑ q using truth tables is provided in the original text and should be included here.)
Conclusion
Logical connectives are fundamental to propositional logic, providing the means to combine simple propositions into complex statements. Understanding their truth tables and equivalences is essential for logical reasoning and analysis.