Problem 48 - 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. You're blushing or sunburned, and you're not sunburned.

Step-by-Step Solution

Answer

The statement is true when 'you're blushing' and 'you're not sunburned'. In terms of symbols, when p is true and q is false, the compound statement (p ∨ q) ∧ ¬q is true.

Step 1: Assign letters to simple statements and Write statement in symbolic form

Let's denote the statement 'you're blushing' as 'p' and 'you're sunburned' as 'q'. The statement can then be written symbolically as: p or q and not q. So the symbolic form is (p ∨ q) ∧ ¬q where ∨ denotes 'or', ∧ denotes 'and' and ¬ is 'not'

Step 2: Construct a truth table

Firstly, list all possible combinations of truth values for statements p and q. Then, evaluate the compound statements (p ∨ q) and ¬q. Finally, put everything together to compute the overall value of statement (p ∨ q) ∧ ¬q. Our truth table will have 4 rows (for every combination of p and q being true and false) and 3 columns (for p, q, and the overall statement).

Step 3: Define the conditions where the statement is true

By examining the truth table, we can determine under what conditions (values of p and q) the compound statement is true.

Step 4: Conclusion

Provide a brief conclusion, based on the results obtained from the truth table and stated conditions under which the compound statement is true.

Key Concepts

Truth Tables

Truth tables are an effective tool used in symbolic logic to determine the truth value of complex logical statements. They help to explore all possible combinations of truth values for the simple statements involved.

In a truth table:

• Each row represents a possible world in which the simple statements can either be true (T) or false (F).
• Columns are used to evaluate the truth values of each part and the entire compound statement.

To illustrate, consider a compound statement involving 'p' for 'you're blushing' and 'q' for 'you're sunburned'. We create rows for all combinations of the truth values for 'p' and 'q'. For example, whether both 'p' and 'q' are true or false, or just one of them is true or false.

This structured approach enables us to see under what conditions the entire statement holds true or false. It's a clear and systematic way to solve logical problems.

Compound Statements

Compound statements combine two or more simple statements using logical connectives. These connectives allow us to express more complex ideas in symbolic logic.

Take the statement 'you're blushing or sunburned, and you're not sunburned'. This can be broken down into simple statements:

• 'p': You're blushing.
• 'q': You're sunburned.

The compound statement involves two main parts:

• 'p or q', implying either is true.
• 'not q', ensuring 'q' is not true.

When translated into symbolic logic, you get (p ∨ q) ∧ ¬q. Using compound statements in symbolic form simplifies the process of determining truth conditions with truth tables. We see the interplay of different components, making it easier to analyze complex scenarios.

Logical Connectives

Logical connectives are the building blocks for forming compound statements in symbolic logic.

The key connectives in the exercise are:

• Disjunction (∨) - This is the 'or' connective. In (p ∨ q), the statement is true if at least one of 'p' or 'q' is true.
• Conjunction (∧) - The 'and' connective. In the statement (p ∨ q) ∧ ¬q, this implies both conditions must be true together.
• Negation (¬) - This is the 'not' connective. It reverses the truth value of a statement. So, ¬q means 'q' is false.

These connectives provide a powerful way to express complex logical relationships. By understanding and applying them, one can break down any verbal statement into its logical components. This not only facilitates precise analysis but also aids in crafting accurate truth tables.