Problem 53 - Thinking Mathematically

Question

Write the negation of each statement. Express each negation in a form such that the symbol \(\sim\) negates only simple statements. \(p \wedge(r \rightarrow \sim s)\)

Step-by-Step Solution

Answer

The negation of the statement \(p \wedge(r \rightarrow \sim s)\) is \(\sim p \vee (r \wedge s)\).

Step 1: Understanding Compound Statement

The given compound statement includes different types of logical operators: conjunction(\(\wedge\)), conditional(\(\rightarrow\)), and negation(\(\sim\)). The statement actually says that 'p and (if r then not s)'.

Step 2: Applying Negation and De Morgan's Law

The first step in negating the compound sentence is to apply the negation operator to the whole sentence. This will give us \(\sim (p \wedge (r \rightarrow \sim s))\). Next, we have to apply De Morgan's law. According to De Morgan's law, the negation of a conjunction is the disjunction of the negations, and the negation of a disjunction is the conjunction of the negations. Therefore, the negation of our compound sentence becomes \(\sim p \vee \sim (r \rightarrow \sim s)\).

Step 3: Simplifying the Conditional Statement

The conditional statement \(r \rightarrow \sim s\) is equivalent to \(\sim r \vee \sim s\). So let's substitute this into our formula. Hence we obtain \(\sim p \vee (\sim (\sim r \vee \sim s))\). Applying De Morgan's law again gives \(\sim p \vee (r \wedge s)\).

Step 4: Final Step

Finally, our compound statement has been simplified to entirely use the symbol \(\sim\) to negate only simple statements. The result is \(\sim p \vee (r \wedge s)\).

Key Concepts

Compound Statement

A compound statement is a logical expression that combines two or more simple statements using logical operators. Each simple statement within a compound statement can be true or false on its own. For example, propositions "p" and "q" could individually be true or false; however, when they are joined, they create a more complex statement.

• The main logical operators used in compound statements are conjunction ( \( \wedge \)), disjunction (\( \vee \)), implication (\( \rightarrow \)), and negation (\( \sim \)).
• Conjunctions combine statements where both must be true for the whole statement to be true.
• Disjunctions are true if at least one component statement is true.
• Implication represents a conditional relationship, expressing "if...then..." scenarios.
• Negation inverts the truth value of a statement: if the original is true, its negation is false, and vice versa.

Our original compound statement \( p \wedge (r \rightarrow \sim s) \) incorporates several of these operators by combining them to make a complex proposition.

De Morgan's Laws

De Morgan's Laws are crucial in the world of logic because they allow us to rewrite logical expressions in different, often simpler forms. These laws deal specifically with negations within compound statements and provide a useful technique for transforming and simplifying complex logical expressions.

• The first law states that the negation of a conjunction is logically equivalent to the disjunction of the negations: \( \sim(p \wedge q) \equiv \sim p \vee \sim q \).
• The second law states that the negation of a disjunction is equivalent to the conjunction of the negations: \( \sim(p \vee q) \equiv \sim p \wedge \sim q \).

These laws are particularly handy when transforming our original expression \( \sim(p \wedge (r \rightarrow \sim s)) \) into \( \sim p \vee \sim(r \rightarrow \sim s) \), further helping us simplify until we reach a form where only simple statements are negated. It gives clarity and structure to seemingly complex logical operations.

Logical Operators

Logical operators are symbols or words used to connect simple statements, allowing them to form compound statements. There are several key operators in logic that each perform specific functions to manipulate truth values in a compound statement.

• Conjunction (\( \wedge \)): Connects two statements and results in true only if both statements are true.
• Disjunction (\( \vee \)): Results in true if at least one of the statements is true.
• Implication (\( \rightarrow \)): Also known as a conditional, it indicates that if the first statement is true, then the second is true.
• Negation (\( \sim \)): Alters the truth value of a statement, making true statements false and vice versa.

These operators are foundational to understanding and constructing compound statements. In our exercise, they serve as the basic mechanism for connecting and manipulating the propositions "p," "r," and "s," ultimately enabling us to explore different configurations and simplifications of the given logical sentence.