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Chapter 6: Problem 10

Prove or disprove the following: (a) \((B \rightarrow A) \leftrightarrow(A \wedge B)\) (b) \((A \wedge B) \rightarrow(A \rightarrow B)\) (c) \((A \wedge B) \rightarrow(A \vee B)\) Note that to disprove a tautology, only one line of a truth table is required.

Short Answer

Expert verified
(a) False, (b) True, (c) True.

Step by step solution

01

Analyze Statement (a)

The given statement is \((B \rightarrow A) \leftrightarrow (A \wedge B)\). We need to prove or disprove this using a truth table or logical reasoning.
02

Disprove Statement (a)

Construct a truth table for \((B \rightarrow A)\), \((A \wedge B)\), and the equivalence \((B \rightarrow A) \leftrightarrow (A \wedge B)\). A counter-example will disprove it:- If \(A = \text{False}, B = \text{True}\), then \(B \rightarrow A = \text{False}, A \wedge B = \text{False}\), making the statement true.- If \(A = \text{True}, B = \text{False}\), then \(B \rightarrow A = \text{True}, A \wedge B = \text{False}\), making the statement false.Thus, \((B \rightarrow A) \leftrightarrow (A \wedge B)\) is false.
03

Analyze Statement (b)

The given statement is \((A \wedge B) \rightarrow (A \rightarrow B)\). We now check its validity with a truth table.
04

Prove Statement (b)

Construct a truth table for \((A \wedge B)\) and \((A \rightarrow B)\):- If \(A = \text{True}, B = \text{True}\), \(A \wedge B = \text{True}, A \rightarrow B = \text{True}\), so the statement is true.- If \(A = \text{True}, B = \text{False}\), \(A \rightarrow B = \text{False}\), and \((A \wedge B) = \text{False}\), the statement is true.- Any case where \(A = \text{False}\) makes \(A \wedge B = \text{False}\) making the implication true since the antecedent is false.Thus, since it's true for all cases, \((A \wedge B) \rightarrow (A \rightarrow B)\) is true.
05

Analyze Statement (c)

The given statement is \((A \wedge B) \rightarrow (A \vee B)\). We'll determine the truth of this statement using logical analysis.
06

Prove Statement (c)

Construct a truth table for \((A \wedge B)\) and \((A \vee B)\):- If \(A = \text{True}, B = \text{True}\), both \(A \wedge B\) and \(A \vee B\) are true.- If either value of \(A\) or \(B\) is false, \(A \wedge B = \text{False}\), making the entire implication true.Therefore, \((A \wedge B) \rightarrow (A \vee B)\) is a tautology.

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Key Concepts

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

Logical Reasoning
Logical reasoning is the process of using a structured and systematic approach to arrive at conclusions based on initial premises or statements. In logic, you often need to prove or disprove statements, which can be translated into logical formulas.

This involves the use of various logical operations such as conjunction (\( \wedge \)), disjunction (\( \vee \)), implication (\( \rightarrow \)), and equivalence (\( \leftrightarrow \)). For instance, a statement like \((A \wedge B) \rightarrow (A \vee B)\) translates to "if both \(A\) and \(B\) are true, then at least one of \(A\) or \(B\) is true."

Logical reasoning helps to analyze such statements by breaking them down to understand each component and their interactions. This requires careful consideration of all possible scenarios, including all potential truth values for each component, to either support or refute the given statements.

To disprove a statement, finding a counter-example 鈥� a scenario in which the statement does not hold true 鈥� is sufficient. Logical reasoning enables an efficient inquiry into the validity of complex propositions, ensuring that all possibilities are accounted for.
Truth Tables
Truth tables are essential tools in logical reasoning and mathematical logic. They allow us to systematically explore all possible truth values of complex logical expressions. By listing all combinations of truth values for the involved variables, we can see how each affects the outcome of the logical statement.

A truth table consists of all possible values for each variable in the expressions, with corresponding true (T) or false (F) values for the expression as a whole. For example, in the truth table for \((B \rightarrow A) \leftrightarrow (A \wedge B)\), each line represents a different combination of truth values for \(A\) and \(B\), showing how they influence the equivalence.

With truth tables, you can:
  • Determine if a logical statement is always true (a tautology), always false (a contradiction), or sometimes true/false (a contingency).
  • Find counter-examples to disprove supposed tautologies.
  • Verify logical equivalences and implications efficiently.
By carefully examining all possible truth values, truth tables offer a clear method to explore complex logical constructs rigorously and thoroughly.
Logical Equivalence
Logical equivalence refers to two statements or expressions having the same truth value in every possible scenario. This is symbolized by \((p \leftrightarrow q)\), meaning that \(p\) is equivalent to \(q\). Logical equivalence is a cornerstone of logical reasoning, allowing us to replace complex expressions with simpler ones that convey the same meaning.

Identifying logical equivalences involves examining expressions for consistent truth values across all possible variable assignments. Two expressions are equivalent if their truth tables produce the same results in each corresponding row.

The understanding of logical equivalence aids in:
  • Simplifying logical expressions to make them easier to work with.
  • Proving theorems by replacing complex parts of logical formulas with equivalent simpler ones.
  • Verifying consistent reasoning in arguments.
In logical reasoning tasks, discovering equivalences is critical for breaking down complicated propositions into manageable components, enabling clearer analysis and processing. It's a powerful tool in both abstract logic and practical problem-solving contexts.

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Most popular questions from this chapter

Bright Homes plc has warehouses in three different locations, \(\mathrm{L}_{1}, \mathrm{~L}_{2}\) and \(\mathrm{L}_{3}\), for making replacement windows. There are three different styles, called 'standard', 'executive' and 'superior': standard units require parts \(B, C\) and \(D\); executive units require parts \(B, C, D\) and \(E\); superior units require parts \(A, B, C\) and \(F\). The parts \(A, B, C, D, E\) and \(F\) are made from components \(a, b, c, d, e, f, g, h\) and \(i\) as follows: \(A\) is made from \(\\{a, b, c\\}\) \(B\) is made from \(\\{c, d, e, f\\}\) \(C\) is made from \(\\{c, e, f, g, h\\}\) \(D\) is made from \(\\{b, e, h\\}\) \(E\) is made from \(\\{c, h, i\\}\) \(F\) is made from \(\\{b, c, f, i\\}\) (a) If the universal set is the set of all components \(\\{a, b, c, d, e, f, g, h, i\\}\), write down the following: $$ \begin{aligned} &\bar{C}, \quad \overline{B \cup C}, \quad \bar{B} \cap \bar{C}, A \cap B \cap D \\ &A \cup F, D \cup(E \cap F),(D \cup E) \cap F \end{aligned} $$ (b) New parts \(B \cup C, C \cup E\) and \(D \cup E \cup F\) are to be made; what are their components? (c) Standard units are made at \(\mathrm{L}_{1}, \mathrm{~L}_{2}\) and \(\mathrm{L}_{3} .\) Executive units are made at \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\) only. Superior units are made at \(\mathrm{L}_{3}\) only. What basic components are needed at each location?

The operation of a machine is monitored on a set of three lamps A, B and C, each of which at any given instant is either 'on' or 'off'. Faulty operation is indicated by each of the following conditions: (a) when both \(\mathrm{A}\) and \(\mathrm{B}\) are off; (b) when all lamps are on; (c) when \(\mathrm{B}\) is on and either \(\mathrm{A}\) is off or \(\mathrm{C}\) is on. Simplify these conditions by describing as concisely as possible the state of the lamps that indicates faulty operation.

Negate the following propositions: (a) Fred is my brother. (b) 12 is an even number. (c) There will be gales next winter. (d) Bridges collapse when design loads are exceeded.

If \(U=\\{1,2,3,4,5,6,7,8,9\\}, A=\\{2,4,6\\}\) \(B=\\{1,3,5,7\\}\) and \(C=\\{2,3,4,7,8\\}\) find the sets (a) \(\overline{A \cup B}\) (b) \(C-A\) (c) \(\bar{C} \cap \bar{B}\)

(a) Draw up truth tables to represent the statements (i) \(p\) is equivalent to \(q\) (ii) \(p\) implies \(q\) (b) Using the algebra of statements, represent the truth of the statements below in tabular form and hence determine whether they are true or false: (i) If \(p\) implies \(q\), and \(r\) implies \(q\), then either \(r\). implies \(p\) or \(p\) implies \(r\). (ii) If \(p\) is equivalent to \(q\), and \(q\) is equivalent to \(r\), then \(p\) implies \(r\).
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