Problem 46 - Thinking Mathematically

Question

a. Write each statement in symbolic form. Assign letters to simple statements that are not negated. b. Construct a truth table for the symbolic statement in part (a). c. Use the truth table to indicate one set of conditions that makes the compound statement true, or state that no such conditions exist. It is not true that I ordered pizza while watching late-night TV and did not gain weight.

Step-by-Step Solution

Answer

The compound statement 'It is not true that I ordered pizza while watching late-night TV and did not gain weight.' is true when 'I ordered pizza and gained weight', when 'I didn't order pizza but did not gain weight' and when 'I didn't order pizza and gained weight'.

Step 1: Write statement in symbolic form

Assign the letter 'P' to represent the statement 'I ordered pizza while watching late-night TV', also assign the letter 'W' to represent 'I did not gain weight'. The given statement can be written symbolically as \( \neg(P \land W) \). 'P' stands for 'I ordered pizza while watching late-night TV' and 'W' stands for 'I did not gain weight'. \( \land \) represents the logical operator AND and \( \neg \) represents NOT.

Step 2: Construct a truth table

A truth table is a table that lists all possible truth values for a compound statement.\n| P | W | P \land W | \( \neg(P \land W) \) |\n| T | T | T | F |\n| T | F | F | T |\n| F | T | F | T |\n| F | F | F | T |\nThe first two columns represent all possible combinations of truth values for P and W. 'T' and 'F' represent 'true' and 'false', respectively. The column 'P \land W' shows the result of P AND W. The final column shows the result of NOT (P and W).

Step 3: Identify conditions that make the statement true

Looking at the last column of the truth table, the conditions that make the original compound statement, 'It is not true that I ordered pizza while watching late-night TV and did not gain weight.', true are when 'P' is true and 'W' is false (I ordered pizza and gained weight), when 'P' is false and 'W' is true (I didn't order pizza but did not gain weight) and when both 'P' and 'W' are false (I didn't order pizza and gained weight).

Key Concepts

Symbolic Logic

Symbolic Logic plays a crucial role in translating everyday language statements into a form that can be analyzed with precision. By converting words into symbols, we can easily apply logical rules to them. In this context, a statement like "It is not true that I ordered pizza while watching late-night TV and did not gain weight" can be transformed into a symbolic form. This translation helps clarify complex statements and enables systematic reasoning. For the given example, the statements "I ordered pizza while watching late-night TV" and "I did not gain weight" are represented as 'P' and 'W' respectively. By using symbols, these statements are then expressed as \( eg(P \land W) \). Here, \( \land \) stands for the logical connector 'AND', while \( eg \) indicates 'NOT' or negation. Such symbolic conversions allow us to simplify compound statements and to use truth tables, which will be discussed further on. This method is common in mathematics, philosophy, and computer science for analyzing logical arguments and improving understanding.

Logical Operators

Logical Operators are the fundamental elements of symbolic logic that allow us to manipulate and combine statements. In our example, logical operators help in forming the truth table. - \( \land \): This is the 'AND' operator. It results in true only if both statements it's connecting are true. Thus, the expression \( P \land W \) is true when both "I ordered pizza while watching late-night TV" and "I did not gain weight" are true.- \( eg \): This is the 'NOT' operator. It simply negates the truth value of the statement it precedes. Therefore, \( eg(P \land W) \) flips the truth value of \( P \land W \), making false outcomes true and vice-versa.Logical operators are integral to constructing and understanding truth tables. They underpin the structure of logic circuits, programming conditions, and much more, highlighting their relevance across disciplines like computer science and mathematics.

Compound Statements

Compound Statements are expressions that combine one or more simple statements using logical operators. They help in evaluating more complex logical expressions, such as our initial statement about ordering pizza and watching TV.To break this down, we first assign simple statements to specific parts of the expression: - 'P' = "I ordered pizza while watching late-night TV."- 'W' = "I did not gain weight."Using these, the compound statement can be expressed as \( eg(P \land W) \). The truth value of the entire expression depends on the truth values of the individual statements 'P' and 'W', combined using the operators. Creating a truth table for this compound statement involves listing all possible truth values for 'P' and 'W' and determining the final result for each scenario. The table helps us to visualize under which conditions the compound statement holds true, aiding in thorough logical analysis and critical thinking. Understanding compound statements is key to logic, as they form the backbone of arguments in various fields.