Problem 2 - Thinking Mathematically

Question

Use Euler diagrams to determine whether each argument is valid or invalid. All physicists arc scientists. All scientists attended college. Therefore, all physicists attended college.

Step-by-Step Solution

Answer

The argument that 'all physicists attended college' is valid, as confirmed by the Euler diagrams.

Step 1: Understand The Statements

The first step is to understand the statements given in the exercise. There are three statements: 'All physicists arc scientists', 'All scientists attended college', and 'Therefore, all physicists attended college.' We must illustrate these relationships using circles in an Euler diagram.

Step 2: Draw the Euler Diagram for the First Statement

For the first statement, 'All physicists are scientists', we draw a large circle to represent all scientists. Inside this circle, we draw a smaller circle to represent physicists. This diagram shows that all physicists are scientists, as the entire physicist circle is inside the scientist circle.

Step 3: Draw the Euler Diagram for the Second Statement

For the second statement, 'All scientists attended college', we draw another larger circle to represent everyone who attended college. The entire scientist circle, which contains the physicist circle, should be within this college circle. This represents that all scientists, and therefore all physicists, attended college.

Step 4: Analysis of the Statements

After drawing the Euler diagrams, we can see that the circle representing physicists is entirely within the circle representing college attendees. This visually confirms the final statement, 'All physicists attended college.' If there were any physicists who did not attend college, they would be represented outside the college circle, which is not the case here.

Key Concepts

Logical Reasoning

Logical reasoning is a fundamental skill that is used to assess whether a series of statements, or a narrative, makes sense or follows logical principles. It is the ability to analyze problems, identify patterns, and come up with rational explanations. When applied to Euler diagrams in logic, it helps in visualizing how different categories relate to each other.

• Analogy: To understand logical reasoning, think of it as putting together a puzzle. Each piece of information is like a puzzle piece that fits with others to complete a picture.
• Inference: Logical reasoning often involves making inferences, which are conclusions reached on the basis of evidence and reasoning.

Improvement to exercises can be made by asking students to explicitly state the logical rules or principles they apply when analyzing the validity of an argument or mapping out an Euler diagram.

Syllogisms

Syllogisms are a type of logical argument that uses deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In the Euler diagram exercise mentioned, the statements presented form a classic syllogism

• Premise 1: All physicists are scientists.
• Premise 2: All scientists attended college.
• Conclusion: Therefore, all physicists attended college.

When we use an Euler diagram to represent this syllogism, we can quickly see whether the conclusion follows logically from the premises. Supplementing textbook exercises with practice in constructing syllogisms can enhance students’ ability to understand and apply logical structures.

Venn Diagrams

Venn diagrams are illustrations used in the branch of mathematics known as set theory. They show all possible logical relations between a collection of different sets. Unlike Euler diagrams, which only show relevant relationships, Venn diagrams represent all possible relations through overlapping circles or other shapes.

Comparing Euler and Venn Diagrams

While Euler diagrams are more straightforward and intuitive, Venn diagrams provide a more rigorous and comprehensive representation, which can be particularly useful for complex logical relationships. Exercises that include the task of translating Euler diagrams to Venn diagrams help students grasp the differences and complement the learning process.

Set Theory

Set theory is the mathematical theory of well-defined collections of objects, which are called sets. It is a fundamental theory underlying many areas of mathematics. Sets can be combined in ways similar to logical propositions.

• Union: Equivalent to logical OR, where elements belong to at least one of the sets.
• Intersection: Equivalent to logical AND, where elements belong to all the sets involved.

Applying Set Theory to Logic

In the context of logical reasoning and Euler diagrams, set theory provides the terminology and principles needed to describe how groups - or sets - of items relate. Teaching set theory concepts directly alongside Euler diagrams can improve students' ability to accurately depict and understand these relationships.