Problem 31 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The resultant truth table will have columns for the simple statements 'p' and 'q', as well as for \(\sim p\), \(\sim q\), \(p \wedge \sim q\) and \(\sim p \vee(p \wedge \sim q)\). For the final solution, the truth values of these statements will depend on the values of 'p' and 'q', evaluated as per the rules of logic.

Step 1: Create the truth table for simple statements

We first create a table with columns for 'p', 'q' and the compound statement. We then fill out the truth values for 'p' and 'q'. The truth values will be True (T) and False (F) and given in such a way that for each possible combination of truth values for 'p' and 'q', there's a row in the table.

Step 2: Evaluate the negation of p and q

We add two new columns to our table, for \(\sim p\) and \(\sim q\). The truth value for these negations will be the opposite of the corresponding simple statement. If 'p' is true, then \(\sim p\) is false, and vice versa. Similarly for \(\sim q\) as well.

Step 3: Evaluate the conjunction of p and negation of q

We add another column to our table for \(p \wedge \sim q\). The truth value of a conjunction is true if and only if both of the conjuncts are true. For each row, we check whether both 'p' and \(\sim q\) are true or not.

Step 4: Evaluate the disjunction of the negation of p and the conjunction evaluated in Step 3

We add the final column to our table for \(\sim p \vee(p \wedge \sim q)\). The truth value of a disjunction is true if at least one of the disjuncts is true. For each row, we check whether either \(\sim p\) or \(p \wedge \sim q\) is true.

Key Concepts

Logical Operators

Logical operators play a crucial role in constructing and evaluating logical statements. They are symbols that help us connect one or more statements to derive new logical conclusions. In truth tables, logical operators help assess every possible truth value for the involved variables. The main logical operators include:

• Negation (\(\sim\)): Reverses the truth value of a statement.
• Conjunction (\(\wedge\)): Yields true only when both component statements are true.
• Disjunction (\(\vee\)): Yields true if at least one component statement is true.

Understanding these operators allows us to perform logical reasoning and construct truth tables effectively, as they dictate how different statements interact with one another based on truth values.

Negation

Negation is one of the simplest yet most powerful logical operations. It involves taking a statement and reversing its truth value. If a statement is originally true, the negation makes it false, and vice versa.

In the context of a truth table, negation is represented by the symbol \(\sim\). For example, if we have a simple statement 'p', its negation is written as \(\sim p\). When constructing a truth table, it is important to add a separate column for \(\sim p\) to show these altered truth values.

Negation is vital when unraveling more complex statements, as it helps us see the flipped perspective of a logical proposition. When evaluating expressions that include negation, always remember its role in converting the true to false and vice versa.

Conjunction

Conjunction involves combining two statements in such a way that the resulting statement is true only if both individual statements are true. The conjunction is symbolized by \(\wedge\), as in the expression \(p \wedge q\).

When laying out a truth table for a conjunction, you'll need to evaluate whether both statements, like 'p' and \(\sim q\), hold true in each scenario.

• If both 'p' and \(\sim q\) are true, then \(p \wedge \sim q\) is true.
• If either or both are false, then \(p \wedge \sim q\) is false.

This logical operator emphasizes the strict condition that both parts must be true, making it more restrictive compared to disjunction.

Disjunction

Disjunction is another core logical operator, represented by the symbol \(\vee\). It connects two statements in a way that the resulting expression is true if at least one of its components is true.

When evaluating disjunctions in a truth table, such as \(\sim p \vee (p \wedge \sim q)\), the focus is on finding scenarios where either part of the disjunction is true:

• If \(\sim p\) is true, or
• If \(p \wedge \sim q\) is true,

then the entire disjunction is true.

Thus, disjunction provides a wider allowance for truthfulness than conjunction, where at least one statement must uphold a truth value, broadening the possibilities of achieving a true outcome in combined statements.