3.6 Euler Diagrams and Syllogistic Arguments 157 U M H H H T T (c) (b) U U M M (a) Figure 3.7 7 Now try Exercise 9 Example 4 Team Mascots Determine whether the following syllogism is valid or invalid. No Knights are Pioneers. Quentin is not a Knight. Quentin is a Pioneer. ∴ Solution The first premise tells us that Knights and Pioneers are disjoint sets. This means that the two sets do not intersect, and the K circle and the P circle should be drawn so they do not overlap. The second premise tells us that Quentin is not a Knight. Thus, Quentin should be placed outside the K circle. Notice from Fig. 3.8 that it is possible to place Quentin outside the K circle without placing him inside the P circle. The Euler diagram shows that the two given premises are satisfied and shows that the conclusion does not necessarily follow from the given premises. Therefore, the argument is invalid. 7 Now try Exercise 13 U K P Q Figure 3.8 Timely Tip Note that in Example 4 if we placed Quentin in the P circle, the argument would appear to be valid. Remember that whenever testing the validity of an argument, always try to show that the argument is invalid . If there is any way of showing that the conclusion does not necessarily follow from the premises, then the argument is invalid. U A B Figure 3.9 Example 5 A Syllogism Involving the Word Some Determine whether the following syllogism is valid or invalid. A B B C A C All ’s are ’s. Some ’s are ’s. Some ’s are ’s. ∴ Solution The premise “All A’s are B’s” is illustrated in Fig. 3.9. The premise “Some B’s are C’s” means that there is at least one B that is a C. We can illustrate this set of premises in four ways, as illustrated in Fig. 3.10. Now try Exercise 19 U B A A A A C C C C (a) U U U B B B (b) (c) (d) Figure 3.10 In all four illustrations, we see that (1) all A’s are B’s and (2) some B’s are C’s. The conclusion is “Some A’s are C’s.” Since at least one of the illustrations, Fig. 3.10(a), shows that the conclusion does not necessarily follow from the given premises, the argument is invalid. 7 Example 6 Fish and Cows Determine whether the following syllogism is valid or invalid. No fish are mammals. All cows are mammals. No fish are cows. ∴ Instructor Resources for Section 3.6 in MyLab Math • Objective-Level Videos 3.6 • PowerPoint Lecture Slides 3.6 • MyLab Exercises and Assignments 3.6
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