Problem 33 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The truth table for the statement \((p \vee q) \wedge(\sim p \vee \sim q)\) will indicate its truth or falsity for each possible combination of truth values for variables \(p\) and \(q\). The final row in the truth table with TT, TF, FT, or FF will signify the logical value (T/F) of the given statement.

Step 1: List All Possible Values for p and q

Begin by listing all possible values for both \(p\) and \(q\). Each can be true (T) or false (F), so there are four possibilities: TT, TF, FT, FF.

Step 2: Calculate Intermediate Values

Calculate the truth values of the intermediate expressions in the given logical operation. Namely, \(p \vee q\) and \(\sim p \vee \sim q\) for each combination of \(p\) and \(q\) values.

Step 3: Compute Final Values

Finally, compute the truth values for the main logical expression \((p \vee q) \wedge(\sim p \vee \sim q)\) using the AND operator on the resulting truth values obtained in the previous step.

Key Concepts

Logical Operations

Logical operations are the backbone of constructing truth tables. In logic, these operations include AND, OR, and NOT, which correspond to specific symbols:

• AND is denoted by \((\wedge)\). It outputs true only when both operands are true.
• OR is denoted by \((\vee)\). It outputs true if at least one operand is true.
• NOT is denoted by \((\sim)\). It inverts the truth value of a statement.

By using these operations, we can transform logical expressions into truth tables, which systematically display the result of various truth values. To master constructing truth tables, understanding how these operations work in combination is crucial. Each operation affects the outcome of the logical expression differently, depending on the input values, thereby creating dynamic interplays that make truth tables a valuable tool in logic.

Truth Values

Truth values play a central role in logic. They represent the two possible states of a proposition: true or false.

• A statement is considered true when it accurately describes reality or a given condition.
• Conversely, a statement is false when it does not accurately reflect reality or the condition.

Logical expressions draw upon these truth values to produce an output for each logical operation. For instance, in a logical expression like \((p \vee q) \wedge(\sim p \vee \sim q)\), each part of the expression speeds through calculations based on these binary truths. Each combination of truth values for \(p\) and \(q\) guides the calculations of the AND, OR, and NOT operations, ultimately allowing the construction of a truth table to illustrate all potential outcomes.

Logical Expressions

Logical expressions are composed of variables and logical operators which combine to form meaningful formulas. In a truth table exercise, you break down these expressions step-by-step. The given expression, \((p \vee q) \wedge(\sim p \vee \sim q)\), involves a combination of OR and AND operations as well as negations.

• The OR operation \((p \vee q)\) checks if either \(p\) or \(q\) is true.
• The NOT operation \((\sim p \vee \sim q)\) takes the opposite of \(p\) and \(q\), applying the OR operation thereafter.
• The AND operation finally combines these results to determine if both internal expressions are true.

Logical expressions illustrate more complex relationships between propositions, and decomposing them is essential to creating accurate truth tables. Mastering these steps not only helps one construct truth tables but also enhances one's understanding of logical analysis.