Problem 38 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

Due to character restrictions, it's not feasible to include the complete truth table here as short answer. Kindly refer to the step-by-step solution for a complete understanding.

Step 1: Understand the logical operators

In this exercise, three logical operators are used: 'and' (\( \wedge \)), 'or' (\( \vee \)), and 'not' (\( \sim \)). Here is how they work: For any propositions p and q, '\( p \wedge q \)' is true if both p and q are true. Otherwise, it is false. '\( p \vee q \)' is false if both p and q are false. Otherwise, it is true. '\( \sim p \)' is true if p is false, and false if p is true.

Step 2: Construct the preliminary truth table

The first step in constructing the truth table is listing out all possible combinations of truth values for the variables p, q, and r. Since there are three variables, there will be eight possible combinations (2^3 = 8).

Step 3: Compute the value of \( \sim q \)

Upon getting the value of q in the preliminary truth table, the value of \( \sim q \) can be computed. This is done by reversing the truth value of q, i.e., '\( \sim q \)' is true if q is false, and '\( \sim q \)' is false if q is true.

Step 4: Compute the value of \( \sim q \wedge r \)

With the truth values of \( \sim q \) and r from the preliminary truth table, the value of \( \sim q \wedge r \) can be computed. If both \( \sim q \) and r are true, then \( \sim q \wedge r \) is true, else, it's false.

Step 5: Compute the value of \( p \vee(\sim q \wedge r) \)

With the truth values of p and \( \sim q \wedge r \) from the preliminary truth table, the value of \( p \vee(\sim q \wedge r) \) can be computed. If either p or \( \sim q \wedge r \) is true, then \( p \vee(\sim q \wedge r) \) is true, else, it's false.

Key Concepts

Logical Operators

Logical operators are the building blocks of propositional logic. They connect propositions to form complex logical statements. These operators include 'and', 'or', and 'not', symbolized as \( \wedge \), \( \vee \), and \( \sim \) respectively. Each operator defines a specific way of combining the truth values of propositions to produce a new truth value.

For example, \( p \wedge q \) represents the logical 'and', which yields true only if both p and q are true independently. The logical 'or', represented as \( p \vee q \), results in true if at least one of p or q is true. Lastly, the logical 'not', denoted by \( \sim p \), inverts the truth value of p - if p is true, then \( \sim p \) is false, and vice versa.

Truth Values

Truth values are the possible outcomes of logical statements or propositions, typically being either true or false. In the binary system of logic, this corresponds to 1 or 0, where 1 indicates 'true' and 0 indicates 'false'.

When constructing truth tables, we explore all possible combinations of truth values for given propositions. This systematic approach helps us deduce the truth value of complex expressions that combine multiple logical operators and propositions. It's essential to cover all possible combinations to understand the behavior of these expressions under different circumstances.

Logical Conjunction

Logical conjunction, signified by the symbol \( \wedge \), is a type of logical operator that combines two propositions with the 'and' function. The result of this conjunction is true only when both propositions are true.

The conjunction is also known as a logical 'and', reflecting the requirement for that both conditions p and q must be met. If we look at it in terms of a truth table: only the combination of true \( \wedge \) true yields true; in all other combinations, the result is false.

Logical Disjunction

Logical disjunction, represented by the symbol \( \vee \), connects two propositions using the logical 'or'. It is true if at least one of the propositions is true. This operator is inclusive, meaning that it does not exclude the possibility that both propositions can be true simultaneously.

For any propositions p and q, the statement \( p \vee q \) will be true if p is true, or q is true, or both are true. The only scenario where the disjunction is false is when both p and q are false.

Logical Negation

Logical negation is the operation of inverting the truth value of a proposition. Denoted by \( \sim \), it turns a true proposition into false and a false proposition into true.

Negation is fundamental in building the truth tables as it allows us to consider the opposite scenarios of a given proposition. It is akin to saying 'it is not the case that...' for any proposition. For instance, if the proposition 'It will rain today,' denoted by p, is true, then \( \sim p \), meaning 'It will not rain today,' is false, and vice versa.