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Chapter 17: Problem 12

Complete the truth table for \((A \vee B) \wedge \sim(A \wedge B)\). $$ \begin{array}{|c|c|c|c|c|c|} \hline A & B & A \vee B & A \wedge B & \sim(A \wedge B) & (A \vee B) \wedge \sim(A \wedge B) \\ \hline \mathrm{T} & \mathrm{T} & & & & \\ \hline \mathrm{T} & \mathrm{F} & & & & \\ \hline \mathrm{F} & \mathrm{T} & & & & \\ \hline \mathrm{F} & \mathrm{F} & & & & \\ \hline \end{array} $$

Short Answer

Expert verified
Final truth values are \(\mathrm{F}, \mathrm{T}, \mathrm{T}, \mathrm{F}.\)

Step by step solution

01

Understanding Disjunction

The disjunction \( A \vee B \) is true if at least one of \( A \) or \( B \) is true. Let's fill this for each row:- When \( A = \mathrm{T} \), \( B = \mathrm{T} \), \( A \vee B \) is True.- When \( A = \mathrm{T} \), \( B = \mathrm{F} \), \( A \vee B \) is True.- When \( A = \mathrm{F} \), \( B = \mathrm{T} \), \( A \vee B \) is True.- When \( A = \mathrm{F} \), \( B = \mathrm{F} \), \( A \vee B \) is False.
02

Understanding Conjunction

The conjunction \( A \wedge B \) is true only if both \( A \) and \( B \) are true. Fill in this for each row:- When \( A = \mathrm{T} \), \( B = \mathrm{T} \), \( A \wedge B \) is True.- When \( A = \mathrm{T} \), \( B = \mathrm{F} \), \( A \wedge B \) is False.- When \( A = \mathrm{F} \), \( B = \mathrm{T} \), \( A \wedge B \) is False.- When \( A = \mathrm{F} \), \( B = \mathrm{F} \), \( A \wedge B \) is False.
03

Negating the Conjunction

The negation \( \sim(A \wedge B) \) is the opposite of \( A \wedge B \). If \( A \wedge B \) is true, \( \sim(A \wedge B) \) is false, and vice versa.- \( A \wedge B = \mathrm{T} \Rightarrow \sim(A \wedge B) = \mathrm{F} \).- \( A \wedge B = \mathrm{F} \Rightarrow \sim(A \wedge B) = \mathrm{T} \).
04

Applying the Compound Function

Now apply the entire expression \((A \vee B) \wedge \sim(A \wedge B)\):- This conjunction is true if both \( A \vee B \) and \( \sim(A \wedge B) \) are true.- For \( (A = \mathrm{T}, B = \mathrm{T}), (A \vee B)\) is \(\mathrm{T}\), \( \sim(A \wedge B)\) is \(\mathrm{F}\), so it is \(\mathrm{F}\).- For \( (A = \mathrm{T}, B = \mathrm{F}), (A \vee B)\) is \(\mathrm{T}\), \( \sim(A \wedge B)\) is \(\mathrm{T}\), so it is \(\mathrm{T}\).- For \( (A = \mathrm{F}, B = \mathrm{T}), (A \vee B)\) is \(\mathrm{T}\), \( \sim(A \wedge B)\) is \(\mathrm{T}\), so it is \(\mathrm{T}\).- For \( (A = \mathrm{F}, B = \mathrm{F}), (A \vee B)\) is \(\mathrm{F}\), \( \sim(A \wedge B)\) is \(\mathrm{T}\), so it is \(\mathrm{F}\).

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Key Concepts

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

Logical Conjunction
Understanding logical conjunction is a foundational aspect of truth tables and logic gates. The logical conjunction, represented by the symbol \( \wedge \), is often referred to as the "AND" operation. It checks if both statements involved are true. This means a conjunction is true if and only if both of the combined conditions are true.

For example, if we have two propositions, \( A \) and \( B \), the conjunction \( A \wedge B \) is true only when both \( A \) and \( B \) are true. Here鈥檚 a simple breakdown to help:
  • \( A = \mathrm{T} \), \( B = \mathrm{T} \): \( A \wedge B = \mathrm{T} \)
  • \( A = \mathrm{T} \), \( B = \mathrm{F} \): \( A \wedge B = \mathrm{F} \)
  • \( A = \mathrm{F} \), \( B = \mathrm{T} \): \( A \wedge B = \mathrm{F} \)
  • \( A = \mathrm{F} \), \( B = \mathrm{F} \): \( A \wedge B = \mathrm{F} \)
This operation is essential in logical decision-making processes and underpins much of computer science and electronic circuit design too.
Logical Disjunction
Logical disjunction is another crucial logical operation, denoted by \( \vee \), and commonly known as the "OR" operation. In contrast to conjunction, disjunction evaluates to true if at least one of the conditions or statements is true. This makes it a bit more flexible than conjunction.

In our truth table exercise, the disjunction \( A \vee B \) reflects these principles:
  • \( A = \mathrm{T} \), \( B = \mathrm{T} \): \( A \vee B = \mathrm{T} \)
  • \( A = \mathrm{T} \), \( B = \mathrm{F} \): \( A \vee B = \mathrm{T} \)
  • \( A = \mathrm{F} \), \( B = \mathrm{T} \): \( A \vee B = \mathrm{T} \)
  • \( A = \mathrm{F} \), \( B = \mathrm{F} \): \( A \vee B = \mathrm{F} \)
This operation is quite natural, resembling real-life scenarios where only one condition needs to be satisfied for an outcome to hold true.
Negation
Negation in logical operations is equivalent to flipping the truth value of a statement. Symbolically represented by \( \sim \), it converts true to false, and false to true.

Let's explore this with an example, focusing on \( \sim (A \wedge B) \). The negation of \( A \wedge B \) essentially makes a true conjunction false and a false conjunction true:
  • If \( A \wedge B = \mathrm{T} \), then \( \sim (A \wedge B) = \mathrm{F} \)
  • If \( A \wedge B = \mathrm{F} \), then \( \sim (A \wedge B) = \mathrm{T} \)
Negation acts as a critical tool in logical reasoning, allowing for the reversal of statements and the construction of more complex logical expressions.
Logical Operations
Logical operations such as conjunction, disjunction, and negation form the backbone of truth tables and logical reasoning. Through these operations, you can analyze and interpret logical expressions, which themselves are the language of formal logic.

In the context of our exercise, the compound logical expression \((A \vee B) \wedge \sim(A \wedge B)\) showcases how different operations are combined to form complex statements. Here's a step-by-step breakdown:
  • First, calculate \( A \vee B \).
  • Next, compute \( A \wedge B \), followed by its negation \( \sim(A \wedge B) \).
  • Finally, combine these results using \(\wedge\).
Proper understanding and application of these logical operations allow for precise troubleshooting and problem-solving within many scientific and everyday contexts.

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Most popular questions from this chapter

For questions \(23-28\), use a Venn diagram or truth table or common form of an argument to decide whether each argument is valid or invalid. If a creature is a chimpanzee, then it is a primate. If a creature is a primate, then it is a mammal. Bobo is a mammal. Therefore, Bobo is a chimpanzee.

For questions \(20-21\), use De Morgan's Laws to rewrite each conjunction as a disjunction, or each disjunction as a conjunction. It is not the case that you need a dated receipt and your credit card to return this item.

For questions \(23-28\), use a Venn diagram or truth table or common form of an argument to decide whether each argument is valid or invalid. Jamie must scrub the toilets or hose down the garbage cans. Jamie refuses to scrub the toilets. Therefore, Jamie will hose down the garbage cans.

For questions \(3-4,\) write the negation of each quantified statement. Everyone failed the quiz today.

For questions \(6-9\), create a truth table for each statement. $$ \sim(\sim A \vee B) $$
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