Problem 32 - Thinking Mathematically

Question

Construct a truth table for the given statement. \(\sim p \wedge(p \vee \sim q)\)

Step-by-Step Solution

Answer

The truth table is as follows:| p | q | \( \sim p \) | \( \sim q \) | \( p \vee \sim q \) | \( \sim p \wedge (p \vee \sim q) \) ||---|---|------|------|---------------|------------------------|| T | T | F | F | T | F || T | F | F | T | T | F || F | T | T | F | F | F || F | F | T | T | T | T |

Step 1: Define all possible combinations for p and q

There are 4 different possibilities for the boolean variables \( p \) and \( q \) as they can either be True (T) or False (F). These are: TT, TF, FT, FF.

Step 2: Determine the NOT operation

The NOT operation (\( \sim \)) inverts the value of the variable that follows it. Therefore, \( \sim p \) and \( \sim q \) will take the opposite value of \( p \) and \( q \) for each of the combinations.

Step 3: Determine the OR operation

Calculate the value of the OR operation (\( \vee \)) for each combination, which involves \( p \) and \( \sim q \). In an OR operation, the result is True if any or both of the inputs are True, else it's False.

Step 4: Determine the AND operation

Finally, calculate the value of the AND operation (\( \wedge \)) for each combination, which involves \( \sim p \) and the result of the OR operation (\( p \vee \sim q \)). In an AND operation, the result is True if both the inputs are True, else it's False.

Key Concepts

Boolean Variables

Boolean variables are the foundation of logic-based systems. They represent values that can be either True or False. In our example, we use the boolean variables \( p \) and \( q \). They can take two possible values each, leading to a combination of outcomes:

• True-True (TT)
• True-False (TF)
• False-True (FT)
• False-False (FF)

These combinations are the basis for constructing a truth table, helping us evaluate logical statements across different scenarios.

Logical Operations

Logical operations are used to manipulate boolean variables and expressions. They include operations like AND, OR, and NOT, which perform basic but powerful functions. In logical computations, these operations transform inputs into meaningful outputs, allowing us to form complex logical expressions.

Negation

Negation, denoted by \( \sim \), is a fundamental logical operation that inverts the value of a boolean variable. If the variable is True, negation makes it False, and vice versa.
For example, given the statement \( \sim p \), if \( p \) is True, \( \sim p \) will be False.
In our scenario, understanding negation is crucial as it helps reverse the state of a variable, influencing the outcome of combined logical operations.

Disjunction

Disjunction is represented by the logical operator \( \vee \), known as OR. It results in True if at least one of the operands is True. If both operands are False, the result is False.

• If both \( p \) or \( \sim q \) (not\( q \)) are True, the disjunction \( p \vee \sim q \) yields True.
• It turns False only if both \( p \) is False and \( q \) is True.

Disjunction allows the inclusion of possibilities, making it an essential part of logical operations.

Conjunction

Conjunction is represented by the operator \( \wedge \), often referred to as AND. It returns True only if both input conditions are True; if either input is False, the operation yields False.

• For the operation \( \sim p \wedge (p \vee \sim q) \), conjunction checks if both \( \sim p \) and the result of \( p \vee \sim q \) are True.
• This narrow condition ensures more restrictive outcomes, which can be beneficial for defining precise logical conditions.

Conjunction is crucial when precise and strict conditions need to be met in logical expressions.