Problem 12
Question
Let \(p\) and q represent the following statements: $$ \begin{aligned} &p: 4+6=10 \\ &q: 5 \times 8=80 \end{aligned} $$ Determine the truth value for each statement. \(\sim p \vee q\)
Step-by-Step Solution
Verified Answer
\(\sim p \vee q\) is false
1Step 1: Evaluate Statement p
Evaluate the truth value of the first statement \(p\), i.e. '4 + 6 = 10'. Solving the equation we see that it indeed equals to 10. Thus statement \(p\) is true.
2Step 2: Evaluate Statement q
Evaluate the truth value of second statement \(q\), i.e. '5 * 8 = 80'. Solving the equation, we see it equals to 40 not 80. Thus statement \(q\) is false.
3Step 3: Calculate \(\sim p \vee q\)
Calculate the truth value of \(\sim p \vee q\). This is the logical disjunction (denoted by \(\vee\)) of the negation (\(\sim\)) of statement \(p\) and statement \(q\). The negation of \(p\) is false because \(p\) is true. Therefore, \(\sim p \vee q\) will yield false OR false, which results in false.
Key Concepts
Truth ValuesLogical OperatorsMathematical Reasoning
Truth Values
Understanding truth values is fundamental in mathematical reasoning and logic. Each statement in logic is assigned a truth value, which can be either true or false. These values help us to perform logical operations and reason about statements or hypotheses.
For instance, in the given exercise, determining the truth value of the statements is the first step. The statement '4 + 6 = 10' is evaluated and found to be true since the sum indeed equals 10. Conversely, the statement '5 × 8 = 80' is evaluated to false because the product is 40, not 80. These evaluations are critical as they form the basis for further logical analysis.
For instance, in the given exercise, determining the truth value of the statements is the first step. The statement '4 + 6 = 10' is evaluated and found to be true since the sum indeed equals 10. Conversely, the statement '5 × 8 = 80' is evaluated to false because the product is 40, not 80. These evaluations are critical as they form the basis for further logical analysis.
Logical Operators
In logic, logical operators are symbols or words used to connect statements or propositions. The most common logical operators include AND (conjunction), OR (disjunction), NOT (negation), IF-THEN (implication), and IF AND ONLY IF (biconditional). Each operator has specific rules for determining the truth value of compound statements they form.
In our exercise, we encounter the logical disjunction operator, denoted by \( \vee \), and the negation operator, denoted by \( \sim \). Disjunction is an OR operation where the compound statement is true if at least one of the individual statements is true. However, in the current context, the disjunction \( \sim p \vee q \) gives us a false outcome since both the negation of p (false) and q (false) are false.
In our exercise, we encounter the logical disjunction operator, denoted by \( \vee \), and the negation operator, denoted by \( \sim \). Disjunction is an OR operation where the compound statement is true if at least one of the individual statements is true. However, in the current context, the disjunction \( \sim p \vee q \) gives us a false outcome since both the negation of p (false) and q (false) are false.
Mathematical Reasoning
The process of mathematical reasoning involves using logic to construct mathematical arguments and reach conclusions based on premises and definitions. Critical thinking and problem-solving are central to this concept. The conclusions must follow logically from the given information.
In the exercise, after establishing the truth values of individual statements, we apply mathematical reasoning to establish the truth value of the compound statement \( \sim p \vee q \). First, we apply the negation to p, switching its truth value. Then, we assess the disjunction of \( \sim p \) and q. We conclude that the compound statement is false, based on logical rules and the previously determined truth values. This process illustrates how mathematical reasoning is used to systematically solve problems and deduce new information from established facts.
In the exercise, after establishing the truth values of individual statements, we apply mathematical reasoning to establish the truth value of the compound statement \( \sim p \vee q \). First, we apply the negation to p, switching its truth value. Then, we assess the disjunction of \( \sim p \) and q. We conclude that the compound statement is false, based on logical rules and the previously determined truth values. This process illustrates how mathematical reasoning is used to systematically solve problems and deduce new information from established facts.
Other exercises in this chapter
Problem 12
Use a truth table to determine whether the two statements are equivalent. \((p \wedge \sim r) \rightarrow q,(\sim p \vee r) \rightarrow \sim q\)
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Construct a truth table for the given statement. \(r \rightarrow(p \vee q)\)
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Let \(p\) and \(q\) represent the following simple statements: \(p\) : This is an alligator. \(q\) : This is a reptile. Write each compound statement in symboli
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Determine whether or not each sentence is a statement. No U.S. president was an only child.
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