Question

Construct a truth table for the statement \(\mathrm{a} \wedge(\mathrm{b} \vee \sim \mathrm{c})\)

Short answer

The truth table for the statement \(\mathrm{a} \wedge (\mathrm{b} \vee \sim \mathrm{c})\) is as follows: | \(\mathrm{a}\) | \(\mathrm{b}\) | \(\mathrm{c}\) | \(\sim \mathrm{c}\) | \(\mathrm{b} \vee \sim \mathrm{c}\) | \(\mathrm{a} \wedge (\mathrm{b} \vee \sim \mathrm{c})\) | |:-----:|:-----:|:-----:|:---------:|:-------------:|:-------------------:| | T | T | T | F | T | T | | T | T | F | T | T | T | | T | F | T | F | F | F | | T | F | F | T | T | T | | F | T | T | F | T | F | | F | T | F | T | T | F | | F | F | T | F | F | F | | F | F | F | T | T | F | This table shows the truth values for the given statement \(\mathrm{a} \wedge (\mathrm{b} \vee \sim \mathrm{c})\) for all possible combinations of \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\).

Step-by-step solution

Determine all possible combinations of truth values for the given variables

Since there are three variables (\(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\)), we will have 8 possible combinations of truth values. List all the possible combinations in a table.

Construct the truth table

Now, we will create a table with columns for \(\mathrm{a}\), \(\mathrm{b}\), \(\mathrm{c}\), \(\sim \mathrm{c}\), \(\mathrm{b} \vee \sim \mathrm{c}\), and the final expression \(\mathrm{a} \wedge (\mathrm{b} \vee \sim \mathrm{c})\). For each row, follow these steps: 1. Assign the truth values of the current combination to \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\). 2. Determine the truth value of \(\sim \mathrm{c}\) using the 'not' operator. 3. Determine the truth value of \(\mathrm{b} \vee \sim \mathrm{c}\) using the 'or' operator. 4. Determine the truth value of the final expression \(\mathrm{a} \wedge (\mathrm{b} \vee \sim \mathrm{c})\) using the 'and' operator. After evaluating all the expressions for their respective truth values, the truth table should look like this: | \(\mathrm{a}\) | \(\mathrm{b}\) | \(\mathrm{c}\) | \(\sim \mathrm{c}\) | \(\mathrm{b} \vee \sim \mathrm{c}\) | \(\mathrm{a} \wedge (\mathrm{b} \vee \sim \mathrm{c})\) | |:-----:|:-----:|:-----:|:---------:|:-------------:|:-------------------:| | T | T | T | F | T | T | | T | T | F | T | T | T | | T | F | T | F | F | F | | T | F | F | T | T | T | | F | T | T | F | T | F | | F | T | F | T | T | F | | F | F | T | F | F | F | | F | F | F | T | T | F | This table shows the truth values for the given statement \(\mathrm{a} \wedge (\mathrm{b} \vee \sim \mathrm{c})\) for all possible combinations of \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\).

Key concepts

Logical Operators

Logical operators are symbols or words used to connect two or more statements in propositional logic, creating compound propositions. In our exercise expression \( \mathrm{a} \wedge(\mathrm{b} \vee \sim \mathrm{c}) \), we see the operators "and" (\( \wedge \)), "or" (\( \vee \)), and "not" (\( \sim \)). Each of these operators has a specific role in evaluating truth statements.
- The "and" operator (\( \wedge \)) signifies that both propositions it connects must be true for the compound statement to be true.- The "or" operator (\( \vee \)) indicates that at least one of the propositions must be true for the compound statement to be true.- The "not" operator (\( \sim \)) negates or reverses the truth value of a given proposition.
These logical operators help us determine the truth of complex statements by breaking them down into simpler parts.

Truth Values

The concept of truth values is central to propositional logic. Every proposition has a truth value. It can be either true (often denoted with T) or false (denoted with F). These values give us a foundation to evaluate logical expressions.
Truth values are also the building blocks of a truth table. In our exercise, we find combinations of truth values for \( \mathrm{a} \), \( \mathrm{b} \), and \( \mathrm{c} \). By systematically assigning true or false values to these variables, we can analyze how the entire expression behaves.
It's important to keep in mind that truth values are distinct and unambiguous, making them a reliable basis for constructing logical arguments and computations.

Propositional Logic

Propositional logic is a branch of logic that involves combining propositions with logical operators to form more complex statements. In essence, propositional logic employs a basic format that allows us to handle more intricate logic problems.
In the exercise, our main statement is \( \mathrm{a} \wedge(\mathrm{b} \vee \sim \mathrm{c}) \). This statement is dissected into simpler propositions \( \mathrm{a} \), \( \mathrm{b} \), and \( \mathrm{c} \), as well as their logical combinations.
Propositional logic helps us not only construct such expressions but is also useful in understanding the interaction between different logical operators, aiding us in predicting the outcomes of various logic-based problems effectively.

Combinations of Truth Values

To construct a truth table, we need to explore all possible combinations of truth values for the given propositions. With three variables \( \mathrm{a} \), \( \mathrm{b} \), and \( \mathrm{c} \), each can independently be true or false.
Given these variables, the total number of combinations is \( 2^3 = 8 \), as each variable has two possibilities (true or false), and they are multiplied together.
Once we have all combinations, we systematically evaluate the entire expression for each configuration. This evaluation considers the negation \( \sim \mathrm{c} \), the disjunction \( \mathrm{b} \vee \sim \mathrm{c} \), and finally, the conjunction \( \mathrm{a} \wedge(\mathrm{b} \vee \sim \mathrm{c}) \). By following each combination, we populate the truth table accurately, providing a complete representation of possible outcomes for the expression.