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Chapter 2: Problem 2

2\. Use truth tables to verify that each of the following is a ggical implication. a) \([(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)\) b) \([(p \rightarrow q) \wedge \neg q] \rightarrow \neg p\) c) \([(p \vee q) \wedge \neg p] \rightarrow q\) d) \([(p \rightarrow r) \wedge(q \rightarrow r)] \rightarrow[(p \vee q) \rightarrow r]\)

Short Answer

Expert verified
After creating and analyzing the truth tables, you'll find that all four given expressions are indeed logical implications. This is because for each expression, in every case where the premise is true, the conclusion is also true.

Step by step solution

01

Construct the truth tables

Create separate truth tables for each of the four logical expressions. Each table will have columns representing the variables \(p\), \(q\), \(r\) (if present), and the logical expression itself. The rows will represent all possible combinations of truth values for the variables, which are true (T) and false (F).
02

Fill in truth values

Fill in the truth values in each row for every logical operation in the expressions. Logical operations should be evaluated in the order of parentheses, negation, conjunction (and), disjunction (or), and implication. Update the truth values in the final column based on the results of these operations.
03

Verify the implications

Check whether each of the given expressions is a logical implication. This is true if every row where the premise is true, the conclusion is also true.

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Key Concepts

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

Logical Implication
Logical implication is a fundamental concept in discrete mathematics and logic. It is represented by the symbol \( \rightarrow \). A logical implication \( A \rightarrow B \) asserts that if proposition \( A \) is true, then proposition \( B \) must also be true. If \( A \) is false, \( B \) can be either true or false; the implication still holds.In truth tables, you check every possible combination of truth values for \( A \) and \( B \). The key point is:
  • If \( A \) is true and \( B \) is true, the implication \( A \rightarrow B \) is true.
  • If \( A \) is true and \( B \) is false, the implication \( A \rightarrow B \) is false.
  • If \( A \) is false, \( A \rightarrow B \) is always true, regardless of \( B \).

Logical implications help in understanding cause-and-effect relationships in logical expressions, forming the basis of proofs and deductions.
Logical Expressions
Logical expressions combine variables and logical operators to form statements that can evaluate to either true or false. In the given problems, expressions are made of variables \( p, q, \) and \( r \), along with operators such as "and" (\( \wedge \)), "or" (\( \vee \)), "not" (\( eg \)), and implications (\( \rightarrow \)).Logical expressions can be complex but can be simplified using truth tables:
  • Negation (\( eg \)): It flips the truth value of a proposition.
  • Conjunction (\( \wedge \)): It is true only if both propositions are true.
  • Disjunction (\( \vee \)): It is true if at least one of the propositions is true.

These expressions are essential in verifying logical implications and are widely used in programming, circuit design, and logical reasoning.
Truth Values
Truth values are the building blocks of logical reasoning. They describe whether a proposition is true (T) or false (F). In the context of truth tables, they allow us to evaluate logical expressions under every possible scenario. Key aspects of truth values include:
  • Binary system: A statement can only be true or false, simplifying the analysis of logical structures.
  • Consistency: Helps maintain logical consistency within reasoning processes.
  • Evaluating expressions: Ensures that logical operations are performed accurately in truth tables.

Understanding truth values is crucial in constructing and analyzing truth tables, which are graphical representations of logical expressions and are vital in verifying logical implications.
Discrete Mathematics
Discrete mathematics deals with structures that are fundamentally separate and distinct. It includes the study of integers, graphs, and logic, among others. Logical implications and expressions are part of this field, laying the groundwork for rigorous logical reasoning. The significance of discrete mathematics includes:
  • Logical foundations: Provides the theoretical basis for logical expressions and programming languages.
  • Decision-making: Assists in making precise and sound decisions in computing and algorithm design.
  • Problem-solving: Offers tools to solve complex problems using logic and reasoning systematically.

Studying discrete mathematics equips students with the necessary skills to tackle various mathematical and computational challenges.

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Most popular questions from this chapter

3\. Use the substitution rules to verify that each of the following is a tautology. (Here \(p, q\), and \(r\) are primitive statements.) a) \([p \vee(q \wedge r)] \vee \neg[p \vee(q \wedge r)]\) b) \([(p \vee q) \rightarrow r] \leftrightarrow[\neg r \rightarrow \neg(p \vee q)]\)

3\. Use the method of exhaustion to show that every even integer between 30 and 58 (including 30 and 58 ) can be written as a sum of at most three perfect squares.

5\. Professor Carlson's class in mechanics is comprised of 29 students of which exactly 1) three physics majors are juniors; 2) two electrical engineering majors are juniors; 3) four mathematics majors are juniors; 4) twelve physics majors are seniors, 5) four electrical engineering majors are seniors; 6) two electrical engineering majors are graduate students; and 7) two mathematics majors are graduate students. Consider the following open statements. \(c(x):\) Student \(x\) is in the class (that is, Professor Carlson's mechanics class as already described). \(j(x): \quad\) Student \(x\) is a junior. \(s(x): \quad\) Student \(x\) is a senior. \(g(x)\) : Student \(x\) is a graduate student. \(p(x): \quad\) Student \(x\) is a physics major. \(e(x): \quad\) Student \(x\) is an electrical engineering major. \(m(x):\) Student \(x\) is a mathematics major. Write each of the following statements in terms of quantifiers and the open statements \(c(x), j(x), s(x), g(x), p(x), e(x)\), and \(m(x)\), and determine whether the given statement is true or false. Here the universe comprises all of the 12,500 students enrolled at the university where Professor Carlson teaches. Furthermore, at this university each student has only one major. a) There is a mathematics major in the class who is a junior. b) There is a senior in the class who is not a mathematics major. c) Every student in the class is majoring in mathematics or physics. d) No graduate student in the class is a physics major. e) Every senior in the class is majoring in either physics or electrical engineering.

10\. Verify that \([p \rightarrow(q \rightarrow r)] \rightarrow[(p \rightarrow q) \rightarrow(p \rightarrow r)]\) is a tautology.

20\. Rewrite each of the following statements in the if-then form. Then write the converse, inverse, and contrapositive of your implication. For each result in parts (a) and (c) give the truth value for the implication and the truth values for its converse, inverse, and contrapositive. [In part (a) "divisibility" requires a remainder of \(0 .]\) a) [The universe comprises all positive integers.] Divisibility by 21 is a sufficient condition for divisibility by \(7 .\) b) [The universe comprises all snakes presently slithering about the jungles of Asia.] Being a cobra is a sufficient condition for a snake to be dangerous. c) [The universe consists of all complex numbers.] For every complex number \(z, z\) being real is necessary for \(z^{2}\) to be real.
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