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For each of these arguments, explain which rules of inference are used for each step. a) 鈥淟inda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket.鈥� b) 鈥淓ach of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year.鈥� c) 鈥淎ll movies produced by John Sayles are wonder-ful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners.鈥� d) 鈥淭here is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre.鈥�

Step by step solution

01

Identifying Given Statements (Part a)

Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket.

02

Rule of Inference (Universal Instantiation)

Apply Universal Instantiation to the second statement: 'Everyone who owns a red convertible has gotten at least one speeding ticket.' This means Linda has gotten a speeding ticket.

03

Conclusion Using Existential Generalization

Since Linda is a student in this class, and Linda has gotten a speeding ticket, use Existential Generalization to conclude that 'Someone in this class has gotten a speeding ticket.'

04

Identifying Given Statements (Part b)

Each of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms.

05

Rule of Inference (Universal Instantiation)

Apply Universal Instantiation to the second statement: 'Every student who has taken a course in discrete mathematics can take a course in algorithms.' Therefore, Melissa, Aaron, Ralph, Veneesha, and Keeshawn can each take a course in algorithms.

06

Conclusion Using Universal Generalization

Since each of the five roommates can take a course in algorithms, conclude that 'All five roommates can take a course in algorithms next year.'

07

Identifying Given Statements (Part c)

All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners.

08

Rule of Inference (Universal Instantiation)

Apply Universal Instantiation to the first statement: 'All movies produced by John Sayles are wonderful.' Therefore, the movie about coal miners produced by John Sayles is wonderful.

09

Conclusion

Conclusively state that 'There is a wonderful movie about coal miners.'

10

Identifying Given Statements (Part d)

There is someone in this class who has been to France. Everyone who goes to France visits the Louvre.

11

Rule of Inference (Existential Instantiation)

Apply Existential Instantiation to the first statement: 'There is someone in this class who has been to France.' This means there is a specific student (call them student X) who has been to France.

12

Rule of Inference (Universal Instantiation)

Apply Universal Instantiation to the second statement: 'Everyone who goes to France visits the Louvre.' Therefore, since student X went to France, student X has visited the Louvre.

13

Conclusion Using Existential Generalization

Since student X is someone in this class and student X has visited the Louvre, conclude that 'Someone in this class has visited the Louvre.'

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Key Concepts

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

Universal Instantiation

In logic, universal instantiation is a rule that allows us to derive a specific instance from a general statement. This comes into play frequently in mathematical proofs and arguments.
For example, in argument (a) from our exercise, we have the general statement: 'Everyone who owns a red convertible has gotten at least one speeding ticket.'
By applying universal instantiation, we derive that Linda, who owns a red convertible, has gotten a speeding ticket. This logical step takes the universal claim and applies it specifically.
Understanding this rule helps in breaking down general statements to use them for individual cases.

Existential Generalization

Existential generalization is another key rule of inference that essentially moves from specific instances to general conclusions.
In argument (a) from our exercise, Linda, a student in the class, has gotten a speeding ticket.
By applying existential generalization, we can state, 'Someone in this class has gotten a speeding ticket.'
This rule allows us to conclude the existence of at least one element that satisfies the condition given.

• Specific instance: Linda has gotten a speeding ticket.
• General conclusion: There exists someone (at least one student) in this class who has gotten a speeding ticket.

This logical step is crucial to move from specific observations to broader generalizations.

Logical Reasoning

Logical reasoning forms the backbone of constructing and deconstructing arguments in discrete mathematics.
It involves a blend of different rules of inference to achieve valid conclusions, ensuring each logical step follows from the previous one without error.
In both argument (b) and (d) in our exercise, logical reasoning helps us conclude that the five roommates can take a course in algorithms and that someone has visited the Louvre.

• Recognize the given statements.
• Apply relevant rules of inference.
• Draw sound conclusions.

Practicing logical reasoning improves your ability to think critically and solve complex problems systematically.

Discrete Mathematics Education

Discrete mathematics deals with mathematical structures that are fundamentally discrete rather than continuous.
A strong grasp of its principles enhances various fields, including computer science, cryptography, and algorithm design.
Education in discrete mathematics often involves studying and understanding concepts like:

• Rules of inference
• Propositional logic
• Combinatorics

In our exercise, understanding concepts like universal instantiation and existential generalization is imperative for mastering discrete mathematics. Engaging with these concepts through exercises strengthens your problem-solving strategies.

Argument Analysis in Mathematics

Argument analysis involves breaking down arguments into their components to understand the logical flow and identify the rules of inference used.
In mathematical contexts, it is crucial to ensure the argument's validity and soundness.
For instance, in argument (c) from our exercise, the steps include:

• Identifying: 'All movies produced by John Sayles are wonderful.'
• Applying Universal Instantiation: 'John Sayles produced a movie about coal miners.'
• Concluding: 'There is a wonderful movie about coal miners.'

By analyzing these steps, students can comprehend how logical connections form valid arguments. This practice is essential for developing airtight mathematical proofs and arguments.