Problem 34 - Thinking Mathematically

Question

Construct a truth table for the given statement. \((p \wedge \sim q) \vee(\sim p \wedge q)\)

Step-by-Step Solution

Answer

The truth table would look like: | \( p \) | \( q \) | \( \sim q \) | \( \sim p \) | \( p \wedge \sim q \) | \( \sim p \wedge q \) | \( (p \wedge \sim q) \vee (\sim p \wedge q) \)--- | --- | --- | --- | --- | --- | --- | ---1. | T | T | F | F | F | F | F2. | T | F | T | F | T | F | T3. | F | T | F | T | F | T | T4. | F | F | T | T | F | F | F

Step 1: Identify the components and setup the truth table

Start by setting up a truth table with columns for \( p \), \( q \), \( \sim q \), \( \sim p \), the individual components \( p \wedge \sim q \) and \( \sim p \wedge q \), and the entire expression. Use T to represent True and F to represent False.

Step 2: Fill in the initial known values

Fill in all combinations of True (T) and False (F) values for \( p \) and \( q \). Also, figure out and fill the values for \( \sim q \) and \( \sim p \), which are negations of \( q \) and \( p \). That means, when \( q \) is True, \( \sim q \) is False, and vice versa. Same goes for \( p \) and \( \sim p \).

Step 3: Compute the expressions

Compute the results for the expressions \( p \wedge \sim q \) and \( \sim p \wedge q \) for each row using the values of \( p \), \( q \), \( \sim p \), and \( \sim q \). If both of the values are true, then the conjunction \( \wedge \) would be true, otherwise, it would be false.

Step 4: Determine the final result

Finally, compute the results for the entire expression \( (p \wedge \sim q) \vee (\sim p \wedge q) \) using the disjunction operator \( \vee \). If either \( (p \wedge \sim q) \) or \( (\sim p \wedge q) \) are true, the result is true. Otherwise, the result is false.

Key Concepts

Logical Operators

Logical operators are essential components in mathematics and computer science. They help to create relationships and comparisons between different elements or statements. The primary logical operators are:

• Negation (\( \sim \)): Flips the truth value of a statement. If a statement is true, negation makes it false, and vice versa.
• Conjunction (\( \wedge \)): Results in true if both statements are true; otherwise, it results in false.
• Disjunction (\( \vee \)): Results in true if at least one of the statements is true; if both are false, it results in false.

These operators form the basis of constructing complex logical statements, like the one given in the exercise. Understanding how they work helps you to analyze and interpret different logical expressions.

Negation

Negation is one of the simplest logical operators. It is represented by the symbol \( \sim \). The purpose of negation is to reverse the truth value of a statement. For example, if you start with a statement \( p \) that is true, \( \sim p \) is false. Conversely, if \( p \) is false, then \( \sim p \) is true.

Using negation is straightforward:

• If \( q \) is True, \( \sim q \) becomes False.
• If \( q \) is False, \( \sim q \) becomes True.

In a truth table, negation allows us to consider the opposite scenario for any given proposition. This is crucial when analyzing statements that depend on not only the original proposition but also its negation.

Conjunction

Conjunction, symbolized by \( \wedge \), combines two statements into one. The conjunction of two statements \( p \) and \( q \) is true only if both \( p \) and \( q \) are true. In all other cases, it is false.

Here is how it works:

• If both \( p \) and \( q \) are True, \( p \wedge q \) is True.
• If either \( p \) or \( q \) is False, \( p \wedge q \) is False.

This operator is used when you need to capture scenarios where multiple conditions must be met simultaneously. In the exercise, expressions like \( p \wedge \sim q \) use conjunction to combine \( p \) with the negation of another proposition, highlighting scenarios where both specific conditions coexist.

Disjunction

Disjunction, denoted by \( \vee \), is used to connect two propositions in such a way that the resulting statement is true if at least one of the propositions is true. Unlike conjunction, a disjunction gives more flexibility as only one condition needs to be satisfied.

Consider how disjunction operates:

• If either \( p \) or \( q \) or both are True, then \( p \vee q \) is True.
• If both \( p \) and \( q \) are False, then \( p \vee q \) is False.

In constructing a truth table, the disjunction operator helps identify when at least one part of a compound statement holds true. The given expression \( (p \wedge \sim q) \vee (\sim p \wedge q) \) employs disjunction to account for circumstances where either conjunction is valid, thereby broadening the possible true outcomes.