Problem 37 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The truth table for the statement \(p \wedge(\sim q \vee r)\) is obtained by determining the truth values for each component of the statement based on the logical connectors and the truth values for the individual propositions. The complete truth table will include all possible combinations of truth values for p, q, and r, and the corresponding truth values for the compound statement.

Step 1: Understand the Symbols

Recognize that the given statement \(p \wedge(\sim q \vee r)\) involves three different propositions: p, q, and r. Likewise, it includes the logical connectors 'and' denoted by \(\wedge\), 'not' denoted by \(\sim\), and 'or' denoted by \(\vee\).

Step 2: Construct the Basic Truth Table

Begin by constructing the truth table for the basics. This includes the three propositions: p, q, and r. Each of these can take on either of two truth values: true (T) or false (F). So, with three variables, there will be \(2^3 = 8\) rows for all the possible combinations of truth values.

Step 3: Determine the Truth Value for \(\sim q\)

Next, calculate the truth values for \(\sim q\). This is the negation of proposition q, meaning its truth value will be the opposite of whatever truth value q has. If q is true, then \(\sim q\) is false, and vice versa.

Step 4: Determine the Truth Value for \((\sim q \vee r)\)

Now determine \((\sim q \vee r)\). This is the 'or' operation between \(\sim q\) and r. According to 'or' operation rule, the statement is true if either \(\sim q\) or r or both are true.

Step 5: Determine the Whole Statement's Truth Value

Finally, get the truth values for the whole statement, \(p \wedge(\sim q \vee r)\). This is the 'and' operation between p and \((\sim q \vee r)\). According the 'and' operation rule, the statement is true if both p and \((\sim q \vee r)\) are true.

Key Concepts

Logical Connectors

In logic, we use specific symbols to represent the relationships between propositions. These symbols are known as logical connectors. They help us form complex statements from simpler ones. A proposition is a statement that can either be true or false, but not both.
The main logical connectors include:

• AND (\(\wedge\)): It is used when both propositions need to be true for the entire statement to be true. Example: "It is sunny AND warm" means it must be both sunny and warm for the overall statement to be true.
• OR (\(\vee\)): It means the statement is true if at least one proposition is true. For example, "It is sunny OR rainy" is true if it's sunny, rainy, or both.
• NOT (\(\sim\)): This flips the truth value of a proposition. If something is true, "NOT" makes it false and vice versa.

By understanding these connectors, you can interpret and construct logical statements and truth tables.

Negation

Negation is a fundamental concept in logic that involves the logical connector "NOT." It reverses the truth value of a given proposition. For example, if a proposition \( q \) is "The sky is blue," its negation \( \sim q \) would be "The sky is NOT blue."
The rules of negation are simple:

• If the original statement is true, the negation is false.
• If the original statement is false, the negation is true.

Negation plays a critical role in logical expressions and truth tables. In our exercise, when you negate \( q \), denoted by \( \sim q \), it changes all true instances of \( q \) to false, and all false instances to true.

Disjunction

Disjunction involves the use of the logical "OR" connector, represented by \( \vee \). This operation combines two propositions and results in a true value if at least one of the propositions is true. This flexibility makes it widely useful in logical expressions.
The truth rules for disjunction are:

• The statement is false only if both propositions are false.
• The statement is true if either or both propositions are true.

In our specific example with \( \sim q \vee r \), the statement will hold true if either \( \sim q \) is true, \( r \) is true, or both are true. The concept of disjunction is pivotal in understanding how multiple conditions can result in a true outcome.

Conjunction

Conjunction uses the logical "AND" connector, denoted by \( \wedge \). This operation mandates that both propositions involved must be true for the overall expression to be true. It is often used when all conditions in a scenario must be satisfied.
The truth rules for conjunction are:

• The statement is false if either proposition is false.
• It is true only when both propositions are true.

For the logical statement \( p \wedge (\sim q \vee r) \) from our exercise, conjunction is used to combine \( p \) with the outcome of \( \sim q \vee r \). Both \( p \) and the result of \( \sim q \vee r \) must be true for the entire expression to hold true. This requirement for simultaneous truth emphasizes the restrictive nature of conjunction in logical statements.