Share
Propositional Logic Truth Tables and Proofs
Published on October 15, 2026
Other
This document covers a series of problems in propositional logic, including constructing truth tables for logical expressions, determining the truth values of compound propositions, finding dual statements, and using logical equivalences to prove tautologies. It also includes applications such as symbolizing natural language statements into logical forms and proving geometric properties using logical reasoning.
Question
Answer
1. 真值表
(1)
|
|
|
|
|
|
|---|---|---|---|---|
| F | F | T | F | F |
| F | T | T | T | T |
| T | F | F | F | T |
| T | T | F | F | T |
(2)
|
|
|
|
|
|---|---|---|---|
| F | T | F | T |
| T | F | T | F |
2. 已知
(1)
(2)
(3)
(2)
(3)
3. 對偶敘述（
(1)
(2)
(2)
4. 真值表（
|
|
|
(1)
|
(2)
|
(3)
|
(4)
|
|---|---|---|---|---|---|
| F | F | F | F | T | T |
| F | T | T | T | F | F |
| T | F | T | T | F | T |
| T | T | F | T | T | F |
5. 複合命題真偽
(1)
6. 證明
故為恆真（重言式）。
7. 符號化
(1)
8. 真值表證明
|
|
|
|
|
|
左式 |
|
右式 |
|---|---|---|---|---|---|---|---|
| F | F | F | T | T | T | F | T |
| F | F | T | T | F | F | T | F |
| F | T | F | T | T | T | F | T |
| F | T | T | T | T | T | T | T |
| T | F | F | F | T | F | T | F |
| T | F | T | F | F | F | T | F |
| T | T | F | T | T | T | T | T |
| T | T | T | T | T | T | T | T |
左、右兩欄完全相同，故等價成立。
9. 命題「
- 逆命題：
- 否命題：
，即 - 逆否命題：
，即
10. 證明
以反證法：假設
與三角形內角和為