Mathematical Logic: Truth Tables & Solutions (HSC 12th)
Solutions and Explanations
Below are the detailed explanations and solutions for Mathematical Logic, specifically focusing on logical connectives and the construction of truth tables, which is a core part of the HSC 12th Standard Mathematics syllabus.
1. Introduction to Logical Connectives
A simple statement is a declarative sentence which is either true or false, but not both simultaneously. We use logical connectives to join simple statements to form compound statements.
- Conjunction (AND): \( \wedge \)
- Disjunction (OR): \( \vee \)
- Negation (NOT): \( \sim \)
- Conditional (Implication): \( \rightarrow \)
- Biconditional (Double Implication): \( \leftrightarrow \)
2. Fundamental Truth Tables
Before solving complex problems, we must understand the standard truth values for each connective.
A. Conjunction (\( p \wedge q \)) and Disjunction (\( p \vee q \))
| p | q | p \( \wedge \) q (AND) | p \( \vee \) q (OR) |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | F |
Note: Conjunction is True only if both are True. Disjunction is False only if both are False.
B. Conditional (\( p \rightarrow q \)) and Biconditional (\( p \leftrightarrow q \))
| p | q | p \( \rightarrow \) q | p \( \leftrightarrow \) q |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | F |
| F | F | T | T |
Note: Implication is False only when Hypothesis (p) is True and Conclusion (q) is False. Biconditional is True when both have the same truth value.
Solved Examples: Constructing Truth Tables
Question 1: Construct the truth table for \( (p \wedge q) \vee \sim p \)
We need columns for \( p \), \( q \), \( \sim p \), \( p \wedge q \), and finally the whole statement.
| p | q | \( \sim p \) | \( p \wedge q \) | \( (p \wedge q) \vee \sim p \) |
|---|---|---|---|---|
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | F | T |
| F | F | T | F | T |
Question 2: Examine whether the statement pattern is a Tautology, Contradiction, or Contingency.
Statement: \( (p \rightarrow q) \leftrightarrow (\sim p \vee q) \)
A Tautology is true in all cases. A Contradiction is false in all cases. A Contingency is a mix.
| p | q | \( \sim p \) | \( p \rightarrow q \) (I) | \( \sim p \vee q \) (II) | (I) \( \leftrightarrow \) (II) |
|---|---|---|---|---|---|
| T | T | F | T | T | T |
| T | F | F | F | F | T |
| F | T | T | T | T | T |
| F | F | T | T | T | T |
Question 3: Three variable Truth Table
Construct the truth table for: \( (p \vee q) \rightarrow r \)
Since there are 3 statements (p, q, r), there will be \( 2^3 = 8 \) rows.
| p | q | r | p \( \vee \) q | \( (p \vee q) \rightarrow r \) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | T | F | T | F |
| T | F | T | T | T |
| T | F | F | T | F |
| F | T | T | T | T |
| F | T | F | T | F |
| F | F | T | F | T |
| F | F | F | F | T |
Important Rules to Remember (Summary)
- Negation: \( \sim T = F \) and \( \sim F = T \)
- Conjunction: T only if T \( \wedge \) T
- Disjunction: F only if F \( \vee \) F
- Conditional: F only if T \( \rightarrow \) F
- Biconditional: T if values match (T \( \leftrightarrow \) T or F \( \leftrightarrow \) F)