Problem 51 - 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. I go to office hours and ask questions, or my professor does not remember me.

Step-by-Step Solution

Answer

The symbolic representation of the statement is \( P \lor \neg Q \). The compound statement is true in all conditions except when 'I do not go to office hours and ask questions' and 'my professor remembers me'.

Step 1: Assigning Symbolic Form

We first identify the simple statements that are not negated: 'I go to office hours and ask questions' and 'my professor does not remember me'. Let's import these statements to: The letter 'P' will represent 'I go to office hours and ask questions' and the letter 'Q' will represent 'my professor does not remember me.' Thus, the compound statement becomes \(P \lor \neg Q\), where \(\lor\) represents 'or', and \(\neg\) represents 'not'.

Step 2: Constructing The Truth Table

A truth table represents all possible truth values of a statement. For this statement, there are four possibilities: 1) both P and Q are true, 2) P is true and Q is false, 3) P is false and Q is true, 4) both P and Q are false. P | Q | \( \neg Q \) | \( P \lor \neg Q \) T | T | F | T T | F | T | T F | T | F | F F | F | T | T .

Step 3: Indicate the Sets of Conditions that Makes the Compound Statement True

From the truth table, we can gather that the compound statement is true in all conditions except when I don't go to office hours and ask questions (P is False) and my professor does remember me (Q is True). Therefore, the compound statement is true as long as I go to office hours and ask questions or if the professor does not remember me.

Key Concepts

Symbolic Logic

Symbolic logic is a branch of mathematics and philosophy that uses symbols to represent logical statements and their relationships. This form of abstraction allows for the creation of precise and unambiguous expressions of logic that are easier to analyze and manipulate than verbal statements. Just like algebra uses symbols for numbers and operations, symbolic logic uses symbols for logical concepts, such as 'and', 'or', 'not', and 'if-then'.

In the given exercise, symbolic logic is employed to simplify the compound statement 'I go to office hours and ask questions, or my professor does not remember me' into a symbolic form. This is done for ease of constructing a truth table and to methodically determine when the statement is true or false. By using symbols like 'P' and 'Q' to represent the individual simpler statements and logical connectors like \(\lor\) for 'or' and \(eg\) for 'not', the complex statement is made straightforward for analysis.

Logical Statements

Logical statements, also known as propositions, are sentences that declare a fact or a condition that can either be true or false, but not both. They are the building blocks of logical reasoning and symbolic logic. A logical statement can be simple, containing a single fact or assertion, or it can be compound, combining multiple statements using logical operations.

In our exercise, two simple logical statements are identified: 'I go to office hours and ask questions' and 'my professor does not remember me'. Each statement by itself can be assigned a truth value: true (T) or false (F). The exercise further combines these simple statements into a compound statement using the logical 'or' operation. Understanding the structure and components of logical statements is crucial because it directly affects their truth values when connected by logical operators. Therefore, breaking down complex sentences into logical statements can greatly simplify the task of evaluating their truth.

Truth Values

Truth values are the foundation of logic, indicating whether a given statement is true (T) or false (F). In symbolic logic, we are mainly concerned with binary truth values as they apply to logical statements. Every logical statement must be either true or false, and when we combine simple statements using logical connectors, their truth values determine the overall truth of the compound statement.

The truth table constructed in our exercise maps out all possible combinations of truth values for the simple statements and shows the resulting truth of the compound statement under each scenario. We see that in the case of \(P \lor eg Q\), the compound statement is true in three out of the four possible truth value combinations. Understanding this concept is critical for students as it helps them assess the validity of arguments and propositions in logical reasoning and in various applications that range from computer science to philosophy.