Problem 51 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The negation of the statement \(p \rightarrow (r \wedge \sim s)\) expressed in a form such that the symbol \(\sim\) negates only simple statements is \( p \wedge (\sim r \vee s) \).

Step 1: Recognize the logical operators

In the expression \(p \rightarrow (r \wedge \sim s)\), \(\rightarrow\) stands for implication and \(\wedge\) signifies conjunction. This expression can be read as: 'If p, then r and not s'.

Step 2: Negate the implication

To negate an implication, recall that \( \sim (p \rightarrow q) = p \wedge \sim q \). Applying this rule, the negation of the whole expression becomes: \( \sim (p \rightarrow (r \wedge \sim s)) = p \wedge \sim (r \wedge \sim s) \).

Step 3: Negate the conjunction

To negate a conjunction, \( \sim (r \wedge \sim s) \), use the rule \( \sim (p \wedge q) = \sim p \vee \sim q \). Hence, the negation becomes: \( \sim (r \wedge \sim s) = \sim r \vee s \).

Step 4: Substitute the simplified negation

Substitute \( \sim r \vee s \) back to \( p \wedge \sim (r \wedge \sim s) \). Hence, the final simplified negation is \( p \wedge (\sim r \vee s) \).

Key Concepts

Implication

An implication is a fundamental concept in logic that forms a connective between two statements. In its simplest form, an implication is expressed as \( p \rightarrow q \), where \( p \) is a hypothesis or premise, and \( q \) is a conclusion. The statement reads as "If \( p \), then \( q \)." This means that whenever \( p \) is true, \( q \) must also be true for the implication to hold. However, when \( p \) is false, the implication is considered true regardless of the truth value of \( q \).
Understanding and negating implications are crucial in logic exercises. To negate an implication \( p \rightarrow q \), one uses the equivalence \( \sim (p \rightarrow q) = p \wedge \sim q \). This translates the statement "It is not true that if \( p \) then \( q \)" into "\( p \) is true, and \( q \) is not true."
When dealing with implications involving more complex statements, like the original \( p \rightarrow (r \wedge \sim s) \), the first step is to apply this rule. This transforms the negation process into a manageable task involving simpler logical operators.

Conjunction

A conjunction is another important logical operator represented by the symbol \( \wedge \). It stands for "and" and connects two statements such that the combined statement is true only if both components are true. So, for a conjunction \( p \wedge q \), the entire statement is only true when both \( p \) and \( q \) are true individually.
Negating a conjunction follows a specific rule that is incredibly useful in simplifying logical statements. This rule is \( \sim (p \wedge q) = \sim p \vee \sim q \). It translates the expression "It is not true that both \( p \) and \( q \) are true" into "either \( p \) is not true, or \( q \) is not true."
In our problem, the conjunction \( r \wedge \sim s \) needed to be negated. By applying the negation rule for conjunctions, \( \sim (r \wedge \sim s) = \sim r \vee s \) was derived. This helped break down the original logical expression into simpler components, allowing for easier manipulation and understanding.

Logical Operators

Logical operators are symbols or words used to connect two or more statements in logic. The most common logical operators include "and" (\( \wedge \)), "or" (\( \vee \)), "not" (\( \sim \)), and "implies" (\( \rightarrow \)). These operators form the foundation of logical expressions and are essential in constructing and understanding logical arguments.

• **And (\( \wedge \)):** Connects two statements in a way that makes the compound statement true only if both parts are true.
• **Or (\( \vee \)):** Another connection method that makes the compound statement true if at least one part is true.
• **Not (\( \sim \)):** Inverts the truth value of a statement. If a statement is true, "not" makes it false, and vice versa.
• **Implies (\( \rightarrow \)):** Connects statements such that if the first is true, the second must also be true for the implication to hold.

In the exercise, these logical operators were employed to build a complex logical statement \( p \rightarrow (r \wedge \sim s) \). Through negation and simplification using known logical laws, each operator played a role in breaking the statement into simpler parts for clearer understanding. Mastery of these operators allows for effective problem-solving and thorough logical reasoning across a wide range of logical tasks.