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Section 3.5

Math in Our World. Section 3.5. Euler Circles. Learning Objectives. Define syllogism . Use Euler circles to determine the validity of an argument. Euler Circles. Euler circles are diagrams similar to Venn diagrams. We will use them to study arguments using four types of statements.

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Section 3.5

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  1. Math in Our World Section 3.5 Euler Circles

  2. Learning Objectives • Define syllogism. • Use Euler circles to determine the validity of an argument.

  3. Euler Circles Euler circles are diagrams similar to Venn diagrams. We will use them to study arguments using four types of statements.

  4. Types of StatementsIllustrated by Euler Circles Type General Form Example Universal affirmative All A is B All chickens have wings. Universal negative No A is B No horses have wings. Particular affirmative Some A is B Some horses are black. Particular negative Some A is not B Some horses are not black. B B A A Universal Affirmative “All A is B” Universal Negative “No A is B”

  5. Types of StatementsIllustrated by Euler Circles Type General Form Example Universal affirmative All A is B All chickens have wings. Universal negative No A is B No horses have wings. Particular affirmative Some A is B Some horses are black. Particular negative Some A is not B Some horses are not black. B B A A x x Particular Affirmative “Some A is B” Particular Negative “Some A is not B”

  6. Syllogism An Argument that consist of two premises and a conclusion is called a syllogism. Premise All cats have four legs. Premise Some cats are black. Conclusion Therefore, some four-legged animals are black. Remember that we are not concerned with whether the conclusion is true or false, but only whether the conclusion logically follows from the premises. If yes, the argument is valid. If no, the argument is invalid.

  7. Determining the Validity of Arguments Euler Circle Method Step 1 Diagram both premises in the same figure. Step 2 If the conclusion is shown in the figure, the argument is valid. Many times the premises can be diagrammed in several ways. If there is even one way in which the diagram contradicts the conclusion, the argument is invalid since the conclusion does not necessarily follow from the premises.

  8. EXAMPLE 1 Using Euler Circles to Determine the Validity of an Argument Use Euler Circles to determine whether the argument is valid. All cats have four legs. Some cats are black. Therefore, some four-legged animals are black.

  9. EXAMPLE 1 Using Euler Circles to Determine the Validity of an Argument SOLUTION The first premise, “All cats have four legs,” is the universal affirmative; the set of cats should be diagrammed as a subset of four-legged animals as shown. The second premise, “Some cats are black,” is the particular affirmative and is shown by placing an x in the intersection of the cats’ circle and the black animals’ circle. The diagram for this premise is drawn on the diagram of the first premise. Four-legged animals Black Cats x Can be drawn two ways. Since there is no other possible diagram, the conclusion “Some four-legged animals are black” is shown to be true.

  10. EXAMPLE 2 Using Euler Circles to Determine the Validity of an Argument Use Euler Circles to determine whether the argument is valid or invalid. Some A is not B. All C is B. ∴ Some A is C.

  11. EXAMPLE 2 Using Euler Circles to Determine the Validity of an Argument SOLUTION The first premise, “Some A is not B,” should be diagrammed as shown. The second premise, “All C is B,” is diagrammed by placing circle C inside circle B. This can be done in several ways, as shown. A A A B B B C C C x x x The third diagram shows that the argument is invalid. It matches both premises, but there are no members of A that are also in C, so it contradicts the conclusion “Some A is C.”

  12. EXAMPLE 3 Using Euler Circles to Determine the Validity of an Argument Use Euler Circles to determine whether the argument is valid or invalid. No criminal is admirable. Some athletes are not criminals. ∴ Some admirable people are athletes.

  13. EXAMPLE 3 Using Euler Circles to Determine the Validity of an Argument SOLUTION The first premise, “No criminal is admirable,” should be diagrammed as shown. The second premise, “Some athletes are not criminals,” can be added in two different ways, as shown. x x Admirable People Admirable People Criminal Criminal Athletes Athletes In the first diagram, the conclusion appears to be valid: some athletes are admirable. But the second diagram doesn’t support that conclusion, so the argument is invalid.

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