Problem 3 - Thinking Mathematically

Question

Use a truth table to determine whether the two statements are equivalent. \(\sim p \rightarrow q, q \rightarrow \sim p\)

Step-by-Step Solution

Answer

After filling out the truth table, it can be seen whether the conditional statements \(\sim p \rightarrow q\) and \(q \rightarrow \sim p\) are logically equivalent. They are logically equivalent only if they have identical truth values across all possible combinations of truth values for their simple statements, p and q.

Step 1: Define Statements and Construct Basic Truth Table

First, define the two simple statements involved: p and q. List all possible truth value combinations for these statements in a truth table. Since there are two statements, there will be \(2^2=4\) rows in the table. Each row represents a different combination of possible truth values for p and q, i.e., (T, T), (T, F), (F, T), and (F, F).

Step 2: Evaluate Negated Statements

Now evaluate the negations in both statements \(\sim p\) and \(\sim q\). For \(\sim p\), if p has a truth value of T, \(\sim p\) will be F, and vice versa. Do the same for \(\sim q\). Record these in the truth table.

Step 3: Evaluate Conditional Statements

Next, evaluate each of the conditional statements \(\sim p \rightarrow q\) and \(q \rightarrow \sim p\). The result of a conditional statement is F only when the first part (antecedent) is true and the second part (consequent) is false; otherwise, it's T. Record these values in the truth table.

Step 4: Determine Logical Equivalence

Finally, determine whether the two statements are logically equivalent. Two statements are logically equivalent if they have the same truth values for all possible combinations of truth values for their simple statements. Consider whether the resulting truth values in the last two columns of the truth table (representing \(\sim p \rightarrow q\) and \(q \rightarrow \sim p\)) are the same for all rows.

Key Concepts

Logical Equivalence

Logical equivalence is a fundamental concept in logic that refers to two statements having the same truth value in every possible scenario. For two statements to be logically equivalent, their truth tables must match exactly, meaning that they are true under the same conditions and false under the same conditions. When analyzing logical equivalence, such as in the exercise involving the statements \(\sim p \rightarrow q\) and \(q \rightarrow \sim p\), we compare their truth values across all possible combinations of truth values for their constituent statements or variables.

In the process of determining logical equivalence, we use a truth table as a visual representation to make the evaluation easier. The truth table shows us whether two complex statements are logically equivalent by lining up their truth values side by side for comparison. A key takeaway for students is that two logically equivalent statements are interchangeable within logical arguments, which makes understanding this concept incredibly valuable for constructing and deconstructing logical expressions.

Conditional Statements

Conditional statements are a type of logical connective that are often expressed in the form \(p \rightarrow q\), which reads \'if p, then q.\