Problem 23 - Discrete Mathematics with Applications

Question

Construct a truth table for each proposition. $$\sim p \vee \sim q$$

Step-by-Step Solution

Answer

The truth table for the proposition \(\sim p \vee \sim q\) is: | p | q | \(\sim p\) | \(\sim q\) | \(\sim p \vee \sim q\) | |-------|-------|---------|---------|---------------------| | True | True | False | False | False | | True | False | False | True | True | | False | True | True | False | True | | False | False | True | True | True |

Step 1: List all possible truth values for p and q

Write the truth values for propositions \(p\) and \(q\). There are four possible combinations: 1. p is True, q is True 2. p is True, q is False 3. p is False, q is True 4. p is False, q is False

Step 2: Find the truth values of ¬p and ¬q

Next, we will find the truth values of \(\sim p\) and \(\sim q\) for each combination: 1. If p is True, then \(\sim p\) is False. 2. If q is True, then \(\sim q\) is False. 3. If p is False, then \(\sim p\) is True. 4. If q is False, then \(\sim q\) is True. Now we will list out the truth values of \(\sim p\) and \(\sim q\) for each combination: 1. p is True, q is True: \(\sim p\) is False, \(\sim q\) is False 2. p is True, q is False: \(\sim p\) is False, \(\sim q\) is True 3. p is False, q is True: \(\sim p\) is True, \(\sim q\) is False 4. p is False, q is False: \(\sim p\) is True, \(\sim q\) is True

Step 3: Find the truth values of ¬p ∨ ¬q

Finally, using the disjunction (\(\vee\)), we will find the truth values for \(\sim p \vee \sim q\): 1. If both \(\sim p\) and \(\sim q\) are False: \(\sim p \vee \sim q\) is False. 2. If at least one of \(\sim p\) and \(\sim q\) is True: \(\sim p \vee \sim q\) is True. Applying this to our list: 1. p is True, q is True: \(\sim p \vee \sim q\) is False. 2. p is True, q is False: \(\sim p \vee \sim q\) is True. 3. p is False, q is True: \(\sim p \vee \sim q\) is True. 4. p is False, q is False: \(\sim p \vee \sim q\) is True. Now, we present the complete truth table: | p | q | \(\sim p\) | \(\sim q\) | \(\sim p \vee \sim q\) | |-------|-------|---------|---------|---------------------| | True | True | False | False | False | | True | False | False | True | True | | False | True | True | False | True | | False | False | True | True | True |

Key Concepts

Propositional Logic

Propositional logic, also known as propositional calculus or logic of propositions, is a branch of logic that deals with the study and analysis of propositions, which are statements that can be either true or false. The purpose of propositional logic is to understand and manipulate the logical relationships between different propositions through various logical operations such as conjunction (AND), disjunction (OR), negation (NOT), implication (IF...THEN), and equivalence.

In propositional logic, these propositions are represented by variables, typically denoted by letters like p and q. These variables can then be used to build more complex expressions by combining them with logical connectives. Truth tables are a powerful tool in propositional logic, as they enable us to outline the results of logical operations for every possible combination of truth values of the variables involved.

Logical Disjunction

Logical disjunction, commonly symbolized as \(\vee\), is an operation in propositional logic that resembles the word 'or' in natural language. When forming a disjunction of two propositions, such as p OR q (denoted as p \(\vee\) q), the resulting compound proposition is true if at least one of the original propositions is true. It only results in a false statement when both original propositions are false.

For instance, if proposition p represents 'It is raining' and proposition q represents 'It is snowing', then the disjunction p \(\vee\) q would be true if either it is raining, it is snowing, or both are happening simultaneously. Contrary to the everyday use of 'or', which often implies exclusivity, the logical disjunction in propositional logic is inclusive, admitting the possibility that both propositions can be true at the same time.

Negation

Negation is a fundamental operation in propositional logic denoted by the symbol \(\sim\) or sometimes as ¬. It is used to reverse the truth value of a proposition. For any proposition p, its negation \(\sim p\) (also read as 'not p') will be true if p is false, and it will be false if p is true.

This unary operation is essential as it allows the construction of complex propositions and is integral to expressing contraries and performing logical argument analysis. For example, if p is the proposition 'The door is open', then \(\sim p\) would represent 'The door is not open' — effectively changing the original statement's truth value. The ability to negate propositions is a key component in establishing logical equivalences and arguments.

Boolean Algebra

Boolean algebra is the area of algebra that deals with the manipulation of truth values, essentially binary values — true or false (or equivalently, 1 or 0). It is named after George Boole, who first defined an algebraic system of logic in the mid-19th century. Boolean algebra is foundational for the field of computer science and digital electronics since it describes the behavior of electrical switches: on or off, true or false.

In Boolean algebra, propositions are evaluated and simplified using a set of operations and laws, such as De Morgan's laws, the commutative law, the associative law, the distributive law, and others. These operations include AND (conjunction), OR (disjunction), NOT (negation), as well as NAND, NOR, XOR (exclusive OR), and XNOR (equivalence). Manipulating Boolean expressions often involves applying these laws to simplify expressions and solve logical propositions.