Problem 2 - A Gentle Introduction to the Art of Mathematics

Question

Complete truth tables for the compound sentences \(A \Longrightarrow B\) and \(\neg A \vee B\).

Step-by-Step Solution

Answer

Truth table for \(A \Longrightarrow B\) is: T, F, T, T and for \(\eg A \vee B\) is: T, F, T, T.

Step 1 - List all possible truth values

List all possible truth values for propositions A and B. Since each proposition can be either true (T) or false (F), there are 4 combinations: 1. A = T, B = T 2. A = T, B = F 3. A = F, B = T 4. A = F, B = F

Step 2 - Determine the truth values for \(A \Longrightarrow B\)

The logical implication \(A \Longrightarrow B\) is false only when A is true and B is false. For the other combinations, it is true. Thus: 1. A = T, B = T: \(A \Longrightarrow B = T\) 2. A = T, B = F: \(A \Longrightarrow B = F\) 3. A = F, B = T: \(A \Longrightarrow B = T\) 4. A = F, B = F: \(A \Longrightarrow B = T\)

Step 3 - Determine the truth values for \(\eg A\)

The negation \(\eg A\) flips the truth value of A. Thus: 1. A = T: \(\eg A = F\) 2. A = T: \(\eg A = F\) 3. A = F: \(\eg A = T\) 4. A = F: \(\eg A = T\)

Step 4 - Determine the truth values for \(\eg A \vee B\)

The disjunction \(\eg A \vee B\) is true if either \(\eg A\) or B is true (or both). Thus: 1. A = T, B = T: \(\eg A \vee B = T\) 2. A = T, B = F: \(\eg A \vee B = F\) 3. A = F, B = T: \(eg A \vee B = T\) 4. A = F, B = F: \(\eg A \vee B = T\)

Step 5 - Combine all steps into truth tables

Combine the results from all previous steps into truth tables for both \(A \Longrightarrow B\) and \(\eg A \vee B\):| A | B | \(A \Longrightarrow B\) | \(\eg A \vee B\) ||-----|-----|-------------------|---------------|| T | T | T | T || T | F | F | F || F | T | T | T || F | F | T | T |

Key Concepts

Logical Implication

Logical implication is a fundamental concept in logic and mathematics. It is expressed as \(A \Longrightarrow B\), read as 'A implies B'. This relationship means that if proposition A is true, then proposition B must also be true for the implication to hold.

Key points about logical implication include:

• If A is true and B is false, \(A \Longrightarrow B\) is false.
• If A is true and B is true, \(A \Longrightarrow B\) is true.
• If A is false, \(A \Longrightarrow B\) is considered true regardless of B's value.

These rules might seem counterintuitive at first, especially the last one. It's because an implication is only false when a true statement implies a false one. In all other cases, including when A is false, the implication is valid.

Understanding these principles makes it easier to complete truth tables and solve logical problems.

Negation

Negation is another core concept of logic. It simply involves flipping the truth value of a proposition. In symbols, the negation of A is written as \(eg A\). If A is true, \(eg A\) is false, and if A is false, \(eg A\) is true.

This flipping operation is straightforward:

• If A is true, \(eg A\) is false.
• If A is false, \(eg A\) is true.

Negation is a basic building block in more complex logical expressions, such as disjunctions and implications.

By understanding negation, you can correctly determine the truth values of more complicated statements.

Disjunction

Disjunction is represented by the symbol \(\vee\) and corresponds to the logical 'or'. The statement \(A \vee B\) asserts that at least one of A or B is true. Importantly, a disjunction is true if either or both of the propositions are true.

Here are the key rules:

• If A is true and B is true, \(A \vee B\) is true.
• If A is true and B is false, \(A \vee B\) is true.
• If A is false and B is true, \(A \vee B\) is true.
• If A is false and B is false, \(A \vee B\) is false.

This logical operator can be combined with negation to form expressions like \(eg A \vee B\). The expression \(eg A \vee B\) is true if either \(eg A\) or B is true.

By mastering disjunction, students can better handle and understand compound logical statements.