Question

Let \(\phi=(p \vee q) \rightarrow(r \wedge \neg s)\). For each of the following interpretations of \(p, q, r,\) and \(s,\) compute \(I(\phi)\) using the truth tables for \(\neg, \vee, \wedge, \rightarrow,\) and \(\leftrightarrow\) (a) \(I(p)=T, I(q)=T, I(r)=T,\) and \(I(s)=F\) (b) \(I(p)=T, I(q)=T, I(r)=F,\) and \(I(s)=F\) (c) \(I(p)=F, I(q)=T, I(r)=T,\) and \(I(s)=T\) (d) \(I(p)=F_{,} I(q)=F_{,} I(r)=T,\) and \(I(s)=T\)

Short answer

(a) T, (b) F, (c) F, (d) T.

Step-by-step solution

Calculate Intermediate Expression for (a)

Given: \( I(p) = T, I(q) = T, I(r) = T, I(s) = F \).First, compute \( p \vee q \):- Both \( p \) and \( q \) are T, so \( p \vee q = T \).Then, compute \( r \wedge eg s \):- \( r = T \).- \( s = F \), so \( eg s = T \).- Thus, \( r \wedge eg s = T \wedge T = T \).

Determine Final Expression for (a)

Using the intermediate expressions:- We have \( T \rightarrow T \), which is T based on the truth table for implication.So, \( I(\phi) = T \) for part (a).

Calculate Intermediate Expression for (b)

Given: \( I(p) = T, I(q) = T, I(r) = F, I(s) = F \).First, compute \( p \vee q \):- \( p = T \) and \( q = T \), so \( p \vee q = T \).Then, calculate \( r \wedge eg s \):- \( r = F \).- \( s = F \), so \( eg s = T \).- Thus, \( r \wedge eg s = F \wedge T = F \).

Determine Final Expression for (b)

Using the intermediate expressions:- We have \( T \rightarrow F \), which is F based on the truth table for implication.So, \( I(\phi) = F \) for part (b).

Calculate Intermediate Expression for (c)

Given: \( I(p) = F, I(q) = T, I(r) = T, I(s) = T \).First, compute \( p \vee q \):- \( p = F \), \( q = T \), hence \( p \vee q = T \).Then, calculate \( r \wedge eg s \):- \( r = T \).- \( s = T \), so \( eg s = F \).- Thus, \( r \wedge eg s = T \wedge F = F \).

Determine Final Expression for (c)

Using the intermediate expressions:- We have \( T \rightarrow F \), which is F by the truth table for implication.So, \( I(\phi) = F \) for part (c).

Calculate Intermediate Expression for (d)

Given: \( I(p) = F, I(q) = F, I(r) = T, I(s) = T \).First, compute \( p \vee q \):- \( p = F \), \( q = F \), hence \( p \vee q = F \).Then, calculate \( r \wedge eg s \):- \( r = T \).- \( s = T \), hence \( eg s = F \).- Thus, \( r \wedge eg s = T \wedge F = F \).

Determine Final Expression for (d)

Using the intermediate expressions:- We have \( F \rightarrow F \), which is T by the truth table for implication.So, \( I(\phi) = T \) for part (d).

Key concepts

Truth Tables

In propositional logic, truth tables are crucial tools. They help us determine the truth value of complex statements by breaking them down into their simpler components. A truth table lists all possible combinations of truth values for variables involved, then shows how these values determine the truth of compound propositions. Each logical connective like "and" (\(\wedge\)), "or" (\(\vee\)), "negation" (\(eg\)), and "implication" (\(\rightarrow\)) has its own truth table. Understanding these tables lets us analyze logical statements systematically. For example, to determine the truth value of an implication \(p \rightarrow q\), we look at possible truth values of \(p\) and \(q\) to see that the statement is only false when \(p\) is true and \(q\) is false.

Propositional Logic

Propositional logic is the branch of logic dealing with propositions and their connectives. It treats propositions as variables that can be either true or false. In a proposition, connectives like "and," "or," "not," and "implies" link simple propositions to form more complex expressions. Let's break it down:

• "And" (\(p \wedge q\)) means both propositions need to be true for the whole expression to be true.
• "Or" (\(p \vee q\)) means at least one of the propositions is true.
• "Not" (\(eg p\)) changes the truth value of a proposition.
• "Implies" (\(p \rightarrow q\)) suggests that if \(p\) is true, \(q\) must also be true.

Each has specific rules for how it affects the truth values in complex statements, outlined in their truth tables. Propositional logic forms the foundation for more advanced logical reasoning.

Implication

The implication, or conditional statement (\(p \rightarrow q\)), plays a vital role in logic. It suggests that if \(p\) is true, then \(q\) must also be true. This type of statement is only false in one condition: when \(p\) is true but \(q\) is false. In all other situations, including when \(p\) is false (regardless of \(q\)), the implication holds true. This might be counterintuitive at first but serves as a fundamental principle in logical deductions. Understanding implication helps us conceptualize dependent conditions in logical expressions, crucial in fields like computer science and mathematics.

Negation

Negation in logic is pretty straightforward yet powerful. It flips the truth value of any given proposition. If a proposition \(p\) is true, its negation \(eg p\) is false, and vice versa. Negation is essential for statements that involve contradictions or exclusions. By using negation, we can directly reverse conditions or assertions within logical arguments, like turning "It is raining" into "It is not raining." This simple transformation changes the meaning and truth value, often serving as a critical component in proving the validity of other logical propositions or in constructing arguments.