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Consider the following argument. All discrete mathematics students can tell a valid argument from an invalid one. All thoughtful people can tell a valid argument from an invalid one. .. All discrete mathematics students are thoughtful. Indicate whether the argument is valid or invalid. O valid O Invalid Support your answer by drawing a diagram. (Select all that apply.) Let D be the set of all discrete mathematics students, T the set of all thoughtful people, and V the set of all people who can tell a valid from an invalid argument.

Consider the following argument. All discrete mathematics students can tell a valid argument from an invalid one. All thoughtful people can tell a valid argument from an invalid one. .. All discrete mathematics students are thoughtful. Indicate whether the argument is valid or invalid. O valid O Invalid Support your answer by drawing a diagram. (Select all that apply.) Let D be the set of all discrete mathematics students, T the set of all thoughtful people, and V the set of all people who can tell a valid from an invalid argument.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Analyzing Argument Validity in Discrete Mathematics** Consider the following argument: 1. All discrete mathematics students can tell a valid argument from an invalid one. 2. All thoughtful people can tell a valid argument from an invalid one. 3. Therefore, all discrete mathematics students are thoughtful. Indicate whether the argument is valid or invalid. - Valid - Invalid (selected) **Supporting Explanation with Diagrams** We can support our answer by drawing a diagram. Let \( D \) be the set of all discrete mathematics students, \( T \) the set of all thoughtful people, and \( V \) the set of all people who can tell a valid argument from an invalid one. ### Diagram Explanation There are two diagrams provided: 1. **Diagram on the Left:** - There are three sets: \(D\), \(T\), and \(V\), with \(D\) and \(T\) inside \(V\). - Both \(D\) and \(T\) are disjoint (do not intersect). - This arrangement suggests that although both \(D\) and \(T\) are subsets of \(V\), there is no direct relationship showing that all elements of \(D\) are also elements of \(T\). 2. **Diagram on the Right:** - Similarly, there are three sets: \(D\), \(T\), and \(V\), where \(D\) and \(T\) are inside \(V\). - In this setup, \(D\) and \(T\) overlap, indicating that there is some intersection between the sets of discrete mathematics students and thoughtful people. - This configuration also shows that not all \(D\) elements are necessarily \(T\) elements. Both diagrams illustrate that the argument is **invalid** because the given premises do not logically lead to the conclusion that all discrete mathematics students are thoughtful. This diagrammatic representation confirms that while the ability to distinguish valid arguments from invalid ones is a quality shared by both discrete mathematics students and thoughtful people, this does not imply all discrete mathematics students are thoughtful people without additional assumptions.
Transcribed Image Text:**Analyzing Argument Validity in Discrete Mathematics** Consider the following argument: 1. All discrete mathematics students can tell a valid argument from an invalid one. 2. All thoughtful people can tell a valid argument from an invalid one. 3. Therefore, all discrete mathematics students are thoughtful. Indicate whether the argument is valid or invalid. - Valid - Invalid (selected) **Supporting Explanation with Diagrams** We can support our answer by drawing a diagram. Let \( D \) be the set of all discrete mathematics students, \( T \) the set of all thoughtful people, and \( V \) the set of all people who can tell a valid argument from an invalid one. ### Diagram Explanation There are two diagrams provided: 1. **Diagram on the Left:** - There are three sets: \(D\), \(T\), and \(V\), with \(D\) and \(T\) inside \(V\). - Both \(D\) and \(T\) are disjoint (do not intersect). - This arrangement suggests that although both \(D\) and \(T\) are subsets of \(V\), there is no direct relationship showing that all elements of \(D\) are also elements of \(T\). 2. **Diagram on the Right:** - Similarly, there are three sets: \(D\), \(T\), and \(V\), where \(D\) and \(T\) are inside \(V\). - In this setup, \(D\) and \(T\) overlap, indicating that there is some intersection between the sets of discrete mathematics students and thoughtful people. - This configuration also shows that not all \(D\) elements are necessarily \(T\) elements. Both diagrams illustrate that the argument is **invalid** because the given premises do not logically lead to the conclusion that all discrete mathematics students are thoughtful. This diagrammatic representation confirms that while the ability to distinguish valid arguments from invalid ones is a quality shared by both discrete mathematics students and thoughtful people, this does not imply all discrete mathematics students are thoughtful people without additional assumptions.
**Transcription and Explanation for Educational Website** --- ### Transcription: ![Concentric Circles Diagram](#) On the left side of the image, there are three concentric circles labeled as follows from the innermost to the outermost circle: - The innermost circle is labeled "D" - The middle circle is labeled "T" - The outermost circle is labeled "V" On the right side of the image, there are three concentric circles as well, but the labeling is different compared to the left side: - The innermost circle is labeled "T" - The middle circle is labeled "D" - The outermost circle is labeled "V" Additionally, there are some symbols surrounding the diagrams: - A small square icon is located below the left circles. - A small square icon with the letter "i" inside is located between the left circles and the right circles. - A small square icon is located below the right circles. - A small icon with the letter "i" inside is located to the right of the right circles. ### Graph Explanation: This image contains two sets of three concentric circles, each set showing a different arrangement of the labels "D", "T", and "V". 1. **Left Diagram:** - The diagram consists of three circles, each inside the other. - The innermost circle is labeled "D". - The middle circle is labeled "T". - The outermost circle is labeled "V". 2. **Right Diagram:** - This diagram also consists of three circles, each inside the other. - The innermost circle is labeled "T". - The middle circle is labeled "D". - The outermost circle is labeled "V". **Potential Interpretation:** - The labels "D", "T", and "V" could represent different tiers or levels within a certain context (e.g., educational structures, organizational hierarchies, classifications in a subject matter). - The difference in the order of labeling between the two diagrams may denote a shift in priority, categorization, or structural changes. ### Conclusion: Understanding these diagrams aids in visualizing hierarchical or tiered structures and how different elements (represented by "D", "T", and "V") relate when arranged in varying orders. For further information or an in-depth discussion regarding the context of these diagrams, please refer to the specific section or chapter of the educational material. ---
Transcribed Image Text:**Transcription and Explanation for Educational Website** --- ### Transcription: ![Concentric Circles Diagram](#) On the left side of the image, there are three concentric circles labeled as follows from the innermost to the outermost circle: - The innermost circle is labeled "D" - The middle circle is labeled "T" - The outermost circle is labeled "V" On the right side of the image, there are three concentric circles as well, but the labeling is different compared to the left side: - The innermost circle is labeled "T" - The middle circle is labeled "D" - The outermost circle is labeled "V" Additionally, there are some symbols surrounding the diagrams: - A small square icon is located below the left circles. - A small square icon with the letter "i" inside is located between the left circles and the right circles. - A small square icon is located below the right circles. - A small icon with the letter "i" inside is located to the right of the right circles. ### Graph Explanation: This image contains two sets of three concentric circles, each set showing a different arrangement of the labels "D", "T", and "V". 1. **Left Diagram:** - The diagram consists of three circles, each inside the other. - The innermost circle is labeled "D". - The middle circle is labeled "T". - The outermost circle is labeled "V". 2. **Right Diagram:** - This diagram also consists of three circles, each inside the other. - The innermost circle is labeled "T". - The middle circle is labeled "D". - The outermost circle is labeled "V". **Potential Interpretation:** - The labels "D", "T", and "V" could represent different tiers or levels within a certain context (e.g., educational structures, organizational hierarchies, classifications in a subject matter). - The difference in the order of labeling between the two diagrams may denote a shift in priority, categorization, or structural changes. ### Conclusion: Understanding these diagrams aids in visualizing hierarchical or tiered structures and how different elements (represented by "D", "T", and "V") relate when arranged in varying orders. For further information or an in-depth discussion regarding the context of these diagrams, please refer to the specific section or chapter of the educational material. ---
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