Problem 26 - Discrete Mathematics with Applications

Question

Construct a truth table for each proposition. $$p \wedge(q \wedge r)$$

Step-by-Step Solution

Answer

The completed truth table for the proposition \(p \wedge (q \wedge r)\) is: | \(p\) | \(q\) | \(r\) | \((q \wedge r)\) | \(p \wedge (q \wedge r)\) | |-----|-----|-----|---------------|------------------------| | T | T | T | T | T | | T | T | F | F | F | | T | F | T | F | F | | T | F | F | F | F | | F | T | T | T | F | | F | T | F | F | F | | F | F | T | F | F | | F | F | F | F | F |

Step 1: List all possible combinations of truth values for \(p\), \(q\), and \(r\)

We have 3 propositions: \(p\), \(q\), and \(r\). Each proposition can either be true (T) or false (F). Therefore, there are \(2^3 = 8\) possible combinations of truth values for these 3 propositions. List all 8 combinations in a table: | \(p\) | \(q\) | \(r\) | |-----|-----|-----| | T | T | T | | T | T | F | | T | F | T | | T | F | F | | F | T | T | | F | T | F | | F | F | T | | F | F | F |

Step 2: Determine the truth value of \((q \wedge r)\) for each row

For each row in the table, determine the truth value of \((q \wedge r)\) using the truth table definition of conjunction (AND): \(\wedge\) is true if and only if both propositions are true. Add a column for \((q \wedge r)\) in the truth table: | \(p\) | \(q\) | \(r\) | \((q \wedge r)\) | |-----|-----|-----|---------------| | T | T | T | T | | T | T | F | F | | T | F | T | F | | T | F | F | F | | F | T | T | T | | F | T | F | F | | F | F | T | F | | F | F | F | F |

Step 3: Determine the truth value of \(p \wedge (q \wedge r)\) for each row

For each row in the table, determine the truth value of the compound proposition \(p \wedge (q \wedge r)\) using the truth table definition of conjunction (AND): \(\wedge\) is true if and only if both propositions are true. Add a column for \(p \wedge (q \wedge r)\) in the truth table: | \(p\) | \(q\) | \(r\) | \((q \wedge r)\) | \(p \wedge (q \wedge r)\) | |-----|-----|-----|---------------|------------------------| | T | T | T | T | T | | T | T | F | F | F | | T | F | T | F | F | | T | F | F | F | F | | F | T | T | T | F | | F | T | F | F | F | | F | F | T | F | F | | F | F | F | F | F | Now, the completed truth table for the proposition \(p \wedge (q \wedge r)\) is as follows: | \(p\) | \(q\) | \(r\) | \((q \wedge r)\) | \(p \wedge (q \wedge r)\) | |-----|-----|-----|---------------|------------------------| | T | T | T | T | T | | T | T | F | F | F | | T | F | T | F | F | | T | F | F | F | F | | F | T | T | T | F | | F | T | F | F | F | | F | F | T | F | F | | F | F | F | F | F |

Key Concepts

Propositions

Propositions are fundamental components in logic that help in forming functional statements. A proposition is basically a declarative sentence that contains a true or false value but not both at the same time. For instance, statements like "The sky is blue." are propositions because it is either true or false based on observation.

• Propositions must be distinct and bear a truth value in logical exercises.


• They are represented in logic as variables, such as \( p, q, \) and \( r \) in our example.


• These variables can be used in various configurations to develop complex expressions, which can further be evaluated with truth tables.

Whenever you encounter logical expressions or problems, identifying and understanding individual propositions is the first crucial step.

Conjunction

The concept of conjunction is a logical operation that combines two propositions and evaluates to true only when both propositions are true. This operation is commonly known as "AND," and it plays a crucial role in constructing compound logical statements.

• The symbol for conjunction is \( \wedge \), so \( p \wedge q \) reads as "\( p \) AND \( q \)."


• This operation is intuitive: it reflects situations where multiple conditions must be satisfied simultaneously.


• In the context of sets, conjunctions reflect the intersection of conditions where all involved propositions agree.

In practical terms, when evaluating conjunction in a truth table, you mark the outcome as true only if every proposition considered in the "AND" expression is true. This strict requirement ensures the precision of logical conclusions.

Logical Operators

Understanding logical operators is essential when dealing with logical expressions. They are special symbols or words used to connect propositions to form more complex logical statements. Each operator has a distinct function and truth condition.

• For example, conjunction (\( \wedge \)) is one of the primary logical operators used for combining statements.


• Other common operators include disjunction (\( \vee \)), which represents "OR," and negation (\( eg \)), which represents "NOT."


• Logical operators are the building blocks of logic used to analyze statements and their relations.

Each operator follows specific rules that set how the truth values of individual propositions influence the truth value of the compound expression. Mastering these operators is crucial for creating and analyzing logical statements accurately in various settings.