3.6 Euler Diagrams and Syllogistic Arguments 159 13. No lawn weeds are flowers. Sedge is not a flower. Sedge is a lawn weed. ∴ Invalid 14. No Cyclones are Longhorns. Zendaya is not a Longhorn. Zendaya is a Cyclone. ∴ Invalid 15. Some mushrooms are poisonous. A morel is a mushroom. A morel is poisonous. ∴ Invalid 16. Some skunks are tame. Pepé is a skunk. Pepé is not tame. ∴ Invalid 17. Some computers are laptops. Some laptops are made by Dell. Some computers are made by Dell. ∴ Invalid 18. Some artists are musicians. Some musicians are politicians. Some artists are politicians. ∴ Invalid 19. Some documentaries are dramas. All dramas are movies. Some documentaries are movies. ∴ Valid 20. Some caterpillars are furry. All furry things are mammals. Some caterpillars are mammals. ∴ Valid 21. No scarecrows are tin men. No tin men are lions. No scarecrows are lions. ∴ Invalid 22. No squirrels are reptiles. No reptiles are birds. No squirrels are birds. ∴ Invalid 23. Some nurses work in pediatrics. Seth works in pediatrics. Seth is a nurse. ∴ Invalid 24. All rainy days are cloudy. Today it is cloudy. Today is a rainy day. ∴ Invalid 25. All Nikons are cameras. A Canon is not a Nikon. A Canon is not a camera. ∴ Invalid 26. All Zebras are pens. A Ticonderoga is not a pen. A Ticonderoga is not a Zebra. ∴ Valid 27. All sweet things taste good. All things that taste good are fattening. All things that are fattening put on pounds. All sweet things put on pounds. ∴ Valid 28. ∴ All cats are dogs. All dogs are sheep. All sheep are pigs. All cats are pigs. Valid Concept/Writing Exercises 29. Can an argument be valid if the conclusion is a false statement? Explain your answer. Yes, if the conclusion necessarily follows from the premises, the argument is valid. 30. Can an argument be invalid if the conclusion is a true statement? Explain. Yes, if the conclusion does not follow from the set of premises, the argument is invalid, even if the conclusion is a true statement. Challenge Problem/Group Activity 31. Sets and Logic Statements in logic can be translated into set statements: For example, p q ∧ is similar to P Q p q ; ∨ > is similar to <P Q; and p q → is equivalent to p q, ∼ ∨ which is similar to < P Q. ′ Euler diagrams can also be used to show that arguments similar to those discussed in Section 3.5 are valid or invalid. Use Euler diagrams to show that the following symbolic argument is invalid. → ∨ ∴∼ p q p q p Research Activity 32. Leonhard Euler Leonhard Euler is considered one of the greatest mathematicians of all time. Do research and write a report on Euler’s life. Include information on his contributions to sets and to logic. Also indicate other areas of mathematics in which he made important contributions. Ravl/Shutterstock
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