Question

Show that \(p \oplus q\) is equivalent to \((p \wedge \neg q) \vee(\neg p \wedge q)\).

Short answer

\(p \oplus q\) is equivalent to \((p \wedge \neg q) \vee (\neg p \wedge q)\) as they both yield identical truth tables.

Step-by-step solution

Understand XOR Operation

The exclusive OR (XOR) operation between two propositions \(p\) and \(q\), denoted as \(p \oplus q\), results in true if exactly one of \(p\) or \(q\) is true, but not both.

Truth Table for XOR

Create a truth table for \(p \oplus q\). The columns should represent the values of \(p\), \(q\), and \(p \oplus q\). For each combination of booleans (True/False) for \(p\) and \(q\), write the corresponding value for \(p \oplus q\).

Setup Components of Equivalent Expression

For the expression \((p \wedge eg q) \vee(eg p \wedge q)\), identify the components: \(p \wedge eg q\) means \(p\) is true and \(q\) is false; \(eg p \wedge q\) means \(p\) is false and \(q\) is true.

Truth Table for Equivalent Expression

Create a truth table for \((p \wedge eg q) \vee (eg p \wedge q)\). For each combination of booleans (True/False) for \(p\) and \(q\), calculate the value of each component \(p \wedge eg q\) and \(eg p \wedge q\), then the OR of both components.

Compare Truth Tables

Compare the truth tables of \(p \oplus q\) and \((p \wedge eg q) \vee (eg p \wedge q)\). Check that both expressions yield the same results for all combinations of \(p\) and \(q\).

Conclude Equivalence

Since the truth tables are identical, it confirms the equivalence of \(p \oplus q\) and \((p \wedge eg q) \vee (eg p \wedge q)\).

Key concepts

XOR operation

In the world of discrete mathematics, the XOR operation stands out for its unique functionality. The XOR, or exclusive OR, between two logical statements, is true if and only if exactly one of the statements is true. To better understand this, consider two propositions, \( p \) and \( q \). When you apply the XOR, expressed as \( p \oplus q \), the result differs from the regular OR operation. Using XOR:

• The result is true if either \( p \) is true, or \( q \) is true, but not both simultaneously.
• It becomes false if both are true or both are false.

This operation is quite handy in digital circuits and error detection processes where such binary outcomes are vital.

truth table

Truth tables are fundamental tools in propositional logic, providing a clear way to visualize the outcomes of logical operations. A truth table lists all possible truth values for the propositions involved and shows the result of the logical operation for each case. Let's consider the XOR operation, \( p \oplus q \). The truth table for this operation can be constructed by considering the four possible combinations of truth values for \( p \) and \( q \):

• When both \( p \) and \( q \) are true, \( p \oplus q \) is false.
• When \( p \) is true and \( q \) is false, \( p \oplus q \) is true.
• When \( p \) is false and \( q \) is true, \( p \oplus q \) is true.
• When both \( p \) and \( q \) are false, \( p \oplus q \) is false.

By constructing such tables, we can systematically analyze and verify the results of logical expressions.

logical equivalence

Logical equivalence can sometimes feel like a puzzle, but it's essentially about finding out if two expressions consistently produce the same result. In propositional logic, two expressions are considered logically equivalent if they yield the same truth values in every possible scenario.Take, for example, the expressions \( p \oplus q \) and \( (p \wedge eg q) \vee (eg p \wedge q) \). The logical equivalence between them can be confirmed by comparing their truth tables:

• Both expressions produce false when \( p \) and \( q \) are both true or both false.
• Both produce true when either \( p \) is true and \( q \) is false, or \( p \) is false and \( q \) is true.

This identical behavior across all combinations proves their logical equivalence, a critical concept for simplifying logical expressions and reasoning.

propositional logic

Propositional logic forms the basis of logical reasoning by facilitating operations with propositions. A proposition is a declarative statement that can either be true or false, and propositional logic deals with how these propositions can be combined and manipulated through logical operations.Key operations in propositional logic include:

• AND (\( \wedge \)) - True if both propositions are true.
• OR (\( \vee \)) - True if at least one proposition is true.
• NOT (\( eg \)) - Inverts the truth value of a proposition.
• XOR (\( \oplus \)) - True if exactly one of the propositions is true.

These operations allow for complex expressions to be built and evaluated. Understanding propositional logic is essential for fields such as computer science, mathematics, and philosophy, where logical reasoning is crucial.