Question

Let proposition \(p\) be \(T\) and proposition \(q\) be \(F\). Find the truth values for the following: (a) \(p \vee q\) (b) \(q \wedge p\) (c) \(\neg p \vee q\) (d) \(p \wedge \neg q\) (c) \(q \rightarrow p\) (f) \(\neg p \rightarrow q\) (g) \(\neg q \rightarrow p\)

Short answer

(a) T, (b) F, (c) F, (d) T, (e) T, (f) T, (g) T.

Step-by-step solution

Evaluate \(p \vee q\)

The disjunction \(p \vee q\) is true if at least one of the propositions is true. Since \(p\) is true \(T\) and \(q\) is false \(F\), \(p \vee q\) evaluates to true \(T\).

Evaluate \(q \wedge p\)

The conjunction \(q \wedge p\) is true only if both propositions are true. Since \(q\) is false \(F\), \(q \wedge p\) evaluates to false \(F\), regardless of the truth of \(p\).

Evaluate \(\neg p \vee q\)

First, determine \(eg p\), which is the negation of \(p\). Since \(p\) is true \(T\), \(eg p\) is false \(F\). The disjunction \(eg p \vee q\) is true if at least one part is true, but since both \(eg p\) and \(q\) are false, \(eg p \vee q\) is false \(F\).

Evaluate \(p \wedge \neg q\)

First, determine \(eg q\), the negation of \(q\). Since \(q\) is false \(F\), \(eg q\) is true \(T\). The conjunction \(p \wedge eg q\) requires both parts to be true, and since both \(p\) and \(eg q\) are true, \(p \wedge eg q\) is true \(T\).

Evaluate \(q \rightarrow p\)

The implication \(q \rightarrow p\) is false only if \(q\) is true and \(p\) is false. Since \(q\) is false, \(q \rightarrow p\) is true \(T\) regardless of \(p\).

Evaluate \(\neg p \rightarrow q\)

First, the negation \(eg p\) is false. In an implication \(eg p \rightarrow q\), if the antecedent (\(eg p\)) is false, the entire implication is true \(T\) regardless of the consequent.

Evaluate \(\neg q \rightarrow p\)

First, \(eg q\) is true \(T\). The implication \(eg q \rightarrow p\) is true if \(p\) is true or if \(p\) is false, since the antecedent \(eg q\) does not lead to a false consequent. Therefore, \(eg q \rightarrow p\) is true \(T\).

Key concepts

Truth Tables

Truth tables are essential tools used in logic, especially in discrete mathematics, to determine the truth values of logical expressions. They are especially helpful in understanding how complex expressions can be broken down into simpler components. A truth table systematically lists all possible combinations of truth values for each proposition involved.

For instance, consider two propositions, \(p\) and \(q\). Each can either be True (T) or False (F). In a truth table, each possible combination of truth values for these propositions is listed, and the resulting truth value of the composed logical expression is evaluated for each combination. This allows us to clearly see how the truth of the components affects the entire expression.

Truth tables are helpful when learning about logical connectives, providing a clear and visual method to understand how various expressions operate in different scenarios.

Logical Connectives

Logical connectives are symbols or words used to connect propositions, forming more complex logical statements. The most common connectives are **AND** \((\land)\), **OR** \((\lor)\), and **NOT** \((eg)\). Each of these operates on one or more propositions to produce a new proposition whose truth value depends on the truth values of the original propositions.

For example:

• The OR connective (\(\lor\)) results in true if at least one of the propositions is true. So, for \(p \lor q\), if either \(p\) or \(q\) is true, the result is true.
• The AND connective (\(\land\)) results in true only if both propositions are true. For \(q \land p\), both \(q\) and \(p\) must be true for the entire expression to be true.
• The NOT connective (\(eg\)) inverts the truth value of a proposition. If \(p\) is true, then \(eg p\) is false, and vice versa.

Understanding these connectives is crucial for working with propositional logic, as they form the basis of constructing and evaluating logical statements.

Propositional Logic

Propositional logic is a branch of logic dealing with propositions, which are statements that are either true or false. In propositional logic, these statements are interpreted as simple or basic propositions that can be connected using logical connectives to form more complex expressions.

The primary goal of propositional logic is to analyze the structure of formed statements and determine their overall truth value. This involves understanding how individual propositions combine, whether through conjunctions (AND), disjunctions (OR), or implications (implies), among others. It's about simplifying complex logical structures to evaluate their outcomes.

Propositional logic is foundational for further studies in logic and computation, feeding into areas such as programming, algorithm design, and even artificial intelligence. The basic principles learned here apply broadly to many disciplines involving problem-solving and logical reasoning.

Logical Implications

Logical implications involve statements of the form \(p \rightarrow q\), read as "if \(p\) then \(q\)". Here, \(p\) is called the antecedent, and \(q\) is the consequent. Logical implication is vital in understanding conditional statements and their truth values.

An implication is a bit tricky at first glance. It's only false when the antecedent \(p\) is true and the consequent \(q\) is false. In all other combinations of truth values, the implication \(p \rightarrow q\) is true. This structure reflects many real-life logical statements, like "if it rains, the street will be wet"—only false if it rains (true \(p\)) and the street is not wet (false \(q\)).

The truth of implications helps to establish logical arguments, determine consequences of certain premises, and is extensively used in proofs and various logical deductions. Understanding them is crucial for logical reasoning and structured thinking across disciplines.