Problem 31 - Discrete Mathematics with Applications

Question

The exclusive disjunetion of two propositions \(p\) and \(q\) is denoted by \(p\) XOR \(q\) . Construct a truth table for \(p\) XOR \(q .\)

Step-by-Step Solution

Answer

The short answer for constructing the truth table for \(p\) XOR \(q\) is: | p | q | p XOR q | |--------|--------|---------| | True | True | False | | True | False | True | | False | True | True | | False | False | False |

Step 1: Identify all possible combinations of truth values for p and q

There are four possible combinations of truth values for two propositions: TT (True-True), TF (True-False), FT (False-True), and FF (False-False).

Step 2: Set up the truth table

Create a table with three columns: one for the truth values of p, one for the truth values of q, and one for the result of p XOR q.

Step 3: Fill in the truth values for p and q

To represent all possible combinations of truth values for p and q, fill the table with the following rows for the p and q columns: TT, TF, FT, and FF.

Step 4: Determine the truth values for p XOR q

For each row in the table, apply the XOR operation to the truth values of p and q, and write the result in the p XOR q column. The XOR operation is true when either p is true or q is true, but not when both are true or both are false.

Step 5: Complete the truth table

After determining the truth values for p XOR q, the final truth table should look as follows: | p | q | p XOR q | |--------|--------|---------| | True | True | False | | True | False | True | | False | True | True | | False | False | False |

Key Concepts

Boolean Algebra

Boolean algebra is a branch of mathematics that deals with variables that have two possible values: true or false. This system of algebra is fundamental to the design and operation of digital circuits and computer logic. It simplifies the analysis and design of electronic circuits and systems, by using binary variables and logical operations such as AND, OR, NOT, and XOR.

At the heart of Boolean algebra is the concept of binary variables that can assume only one of two possible states, which are typically labeled as 1 (true) and 0 (false). Operations on these variables are defined by truth tables that show the outcome for every possible combination of inputs. The exclusive or, or XOR, operation is one such binary operation that outputs true only when the inputs differ. The ability to manipulate these true and false values using logical operations forms the backbone of modern computing and digital electronics.

Logical Connectives

Logical connectives, also known as logical operators, are symbols or words used to connect clauses or statements to form a complex proposition in logic. They include AND (conjunction), OR (disjunction), NOT (negation), and XOR (exclusive disjunction). Each of these operators has a specific set of rules it follows, which can be represented in a truth table.

For example, the AND operator is true only when both operands are true. The OR operator is true when at least one operand is true. NOT simply inverts the truth value of its operand. XOR stands out by being true only when there is a truth value difference between its two operands—it is false when both are the same. These operations allow the building of more complex logical statements from basic ones, which is crucial in fields such as mathematics, computer science, and philosophy.

Propositional Logic

Propositional logic, also known as sentential logic, is a widely studied area in the field of logic that focuses on manipulating and evaluating sentences that can be true or false, known as propositions. It forms the foundation for reasoning about the truth or falsity of statements in logical systems.

Within propositional logic, a sentence is constructed from simple propositions, such as 'p' and 'q', using logical connectives. The complexity of these statements can be assessed using truth tables, which provide a systematic method to enumerate and evaluate the possible combinations of truth values for the propositions involved. This evaluation is crucial for understanding logical equivalences, performing deductive reasoning, and application in computer logic design. By examining each possible configuration, one can determine the overall truth value of complex statements, ensuring the reliable operation of logical schemes and electronic devices built upon these principles.