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Chapter 1: Problem 30

Create a complete truth table for each statement. If \((B\) or \(C)\), then \((A\) and \(B)\)

Short Answer

Expert verified
The truth table has 8 rows showing each possible truth value combination for \((B \lor C) \rightarrow (A \land B)\).

Step by step solution

01

Identify the Components

We need to analyze the logical statement: \((B \lor C) \rightarrow (A \land B)\). It involves the logical operators OR (\(\lor\)), AND (\(\land\)), and IF-THEN (\(\rightarrow\)). Notice the variables are \(A\), \(B\), and \(C\).
02

Determine Possible Truth Values

Since the statement involves three variables (\(A\), \(B\), and \(C\)), each variable can be either true (T) or false (F). Thus, there are \(2^3 = 8\) combinations of truth values for the three variables.
03

Create the Initial Truth Table

List all possible combinations of truth values for \(A\), \(B\), and \(C\) in a table format. The truth table should have 8 rows (one for each combination) and at least 4 columns: \(A\), \(B\), \(C\), and the main expression.
04

Add Intermediate Columns

To evaluate \((B \lor C)\) and \((A \land B)\), add two more columns to the table: one for \((B \lor C)\) and another for \((A \land B)\).
05

Evaluate \((B \lor C)\)

Fill in the column for \((B \lor C)\): For each row, the result is T if at least one of \(B\) or \(C\) is T, and F otherwise.
06

Evaluate \((A \land B)\)

Fill in the column for \((A \land B)\): For each row, the result is T only if both \(A\) and \(B\) are T, and F otherwise.
07

Evaluate the Entire Expression \((B \lor C) \rightarrow (A \land B)\)

Fill in the column for the main expression. Using the implication rule, the result of \((B \lor C) \rightarrow (A \land B)\) is F only if \((B \lor C)\) is true and \((A \land B)\) is false. Otherwise, it is T.
08

Fill in the Completed Truth Table

Organize the table with columns: \(A\), \(B\), \(C\), \(B \lor C\), \(A \land B\), and \((B \lor C) \rightarrow (A \land B)\). Ensure all truth values across the rows adhere to logical outcomes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logical Operators
Logical operators are symbolic representations used in logical expressions to perform specific logical functions. Some common logical operators include AND (\( \land \)), OR (\( \lor \)), NOT (\( \lnot \)), and IF-THEN (\( \rightarrow \)). Each of these operators performs a distinct function when applied to truth values.
  • **AND Operator**: Denoted as \( \land \) and it returns true only if both operands are true. For instance, if \( A \) is true and \( B \) is true, then \( A \land B \) is true. Otherwise, it is false.
  • **OR Operator**: Denoted as \( \lor \) and it returns true if at least one operand is true. So, \( A \lor B \) is true if either \( A \) or \( B \) is true or if both are true.
  • **IF-THEN Operator**: Denoted as \( \rightarrow \) which is a conditional operator. It returns false only if the antecedent (the first part) is true and the consequent (the second part) is false. Otherwise, it returns true.
Understanding these operators is crucial when creating a truth table because they dictate how different truth values interact and result in a final outcome.
Logical Expressions
A logical expression is a combination of logical operators and variables that can be evaluated to a single truth value. In the context of logical expressions, we use variables such as \( A \), \( B \), and \( C \), which can either be true or false.
  • Logical expressions follow rules of logical equivalences and can be represented in different forms, such as disjunctive normal form and conjunctive normal form.
  • The expression \((B \lor C) \rightarrow (A \land B)\) combines multiple operators and requires intermediate computations to determine the overall truth value.
  • The evaluation of such expressions generally starts with simpler operations, leading towards the complete evaluation of the compound statement.
To fully comprehend logical expressions, it's essential to understand the precedence of operators (i.e., which operations to perform first) and how they are combined to form more complex expressions.
Truth Values
Truth values are the building blocks of logic. They represent the possible states of a logical expression: true (T) or false (F). When constructing a truth table, various combinations of truth values are analyzed to determine the outcome of a logical expression.
  • Each logical variable can be one of two states, making logical systems binary.
  • The truth value of an expression is determined by the truth values assigned to its variables and the logical operators used.
  • In a truth table for an expression like \((B \lor C) \rightarrow (A \land B)\), every possible combination of \(A\), \(B\), and \(C\) is evaluated to decide the truth value of the entire expression.
Understanding truth values is key to interpreting the result of logical expressions correctly. It helps to see clearly how each part of the expression contributes to the end result.

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Most popular questions from this chapter

For each of the following, draw the appropriate illustration of sets (Subset, Disjoint or Overlapping). Then put an \(\mathrm{X}\) to represent the subject of the conclusion or put two question marks to illustrate the subject could into two locations. Finally, state if the argument is valid and whether it is sound. Premise: Some entering freshmen have to take a placement test. Premise: Juan is an entering freshman. Conclusion: Juan has to take a placement test.

For each situation, decide whether the "or" is most likely exclusive or inclusive. a. I will wear a sweater or a jacket. b. My next vacation will be on the Oregon Coast or Mount Hood.

For each argument, draw the appropriate illustration of sets (Subset, Disjoint or Overlapping). Then put an X to represent the subject of the conclusion. Alternatively, use two question marks to illustrate the subject could fit into two locations. Finally, state whether the conclusion is valid and whether it is sound. Some students play sports. Javeer plays soccer. Conclusion: Javeer is a student.

Express in the form of if \(p,\) then \(q .\) Identify \(\mathrm{p}\) and \(\mathrm{q}\) a. Doing homework helps increase your grade in class. b. Riding public transportation will help you save money. c. Squirrels, bury their food. d. You will get sick if you eat too much candy. e. Go to the doctor if you think you have the flu.

Write the negation of each proposition. a. I take public transportation to get to class. b. I went to a movie on Friday. c. I don't want to go golfing today. d. I love watching basketball. e. Breylynn's favorite color is green f. Mirriam is a theater major.
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