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Deductive Reasoning and Laws of Logic: Examples, Strategies, and Practice

Learn about deductive reasoning and the laws of logic with real-world examples and practice problems. Understand the law of detachment, law of syllogism, and how to judge conclusions using Venn diagrams.

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Deductive Reasoning and Laws of Logic: Examples, Strategies, and Practice

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  1. LESSON 2–4 Deductive Reasoning

  2. Five-Minute Check (over Lesson 2–3) TEKS Then/Now New Vocabulary Example 1: Real-World Example: Inductive and Deductive Reasoning Key Concept: Law of Detachment Example 2: Law of Detachment Example 3: Judge Conclusions Using Venn Diagrams Key Concept: Law of Syllogism Example 4: Law of Syllogism Example 5: Apply Laws of Deductive Reasoning Lesson Menu

  3. Identify the hypothesis and conclusion. If 6x – 5 = 19, then x = 4. A. Hypothesis: 6x – 5 = 19 Conclusion: x = 4 B. Hypothesis: 6x – 5 = 19Conclusion: x  4 C. Hypothesis: x = 4Conclusion: 6x – 5 = 19 D. Hypothesis: 6x – 5  19Conclusion: x = 4 5-Minute Check 1

  4. Identify the hypothesis and conclusion. A polygon is a hexagon if it has six sides. A. Hypothesis: The polygon is not a hexagon.Conclusion: A polygon has six sides. B. Hypothesis: The polygon is a hexagon.Conclusion: A polygon has six sides. C. Hypothesis: A polygon has six sides.Conclusion: The polygon is a hexagon. D. Hypothesis: A polygon does not have six sides.Conclusion: The polygon is a hexagon. 5-Minute Check 2

  5. Which choice shows the statement in if-then form?Exercise makes you healthier. A. If you exercise, then you will be healthier. B. If you are healthy, then you exercise. C. If you don’t exercise, you won’t be healthier. D. If you don’t exercise, then you won’t be healthier. 5-Minute Check 3

  6. Which choice shows the statement in if-then form? Squares have 4 sides. A. If a figure has 4 sides, then it is a square. B. If a figure is a square, then it has 4 sides. C. A figure is not a square if it does not have 4 sides. D. If a figure is not a square, then it does not have 4 sides. 5-Minute Check 4

  7. Given the statement, “If you live in Miami, then you live in Florida,” which of the following is true? A. the negation of the statement B. the inverse of the statement C. the converse of the statement D. the contrapositive of the statement 5-Minute Check 5

  8. Targeted TEKS Preparation for G.4(B) Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between biconditional statement and a true conditional statement with a true converse. Mathematical Processes G.1(F), G.1(G) TEKS

  9. You used inductive reasoning to analyze patterns and make conjectures. • Use the Law of Detachment. • Use the Law of Syllogism. Then/Now

  10. deductive reasoning • valid • Law of Detachment • Law of Syllogism Vocabulary

  11. Inductive and Deductive Reasoning A. WEATHER Determine whether the conclusion is based on inductive or deductive reasoning. In Miguel’s town, the month of April has had the most rain for the past 5 years. He thinks that April will have the most rain this year. Answer: Miguel’s conclusion is based on a pattern of observation, so he is using inductive reasoning. Example 1

  12. Inductive and Deductive Reasoning B. WEATHER Determine whether the conclusion is based on inductive or deductive reasoning. Sandra learned that if it is cloudy at night it will not be as cold in the morning than if there are no clouds at night. Sandra knows it will be cloudy tonight, so she believes it will not be cold tomorrow morning. Answer: Sandra is using facts that she has learned about clouds and temperature, so she is using deductive reasoning. Example 1

  13. A.Determine whether the conclusion is based on inductive or deductive reasoning. Macy’s mother orders pizza for dinner every Thursday. Today is Thursday. Macy concludes that she will have pizza for dinner tonight. A. inductive B. deductive Example 1

  14. B.Determine whether the conclusion is based on inductive or deductive reasoning.The library charges $0.25 per day for overdue books. Kyle returns a book that is 3 days overdue. Kyle concludes that he will be charged a $0.75 fine. A. inductive B. deductive Example 1

  15. Concept

  16. Law of Detachment Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning.Given: If a figure is a square, then it is a parallelogram. The figure is a parallelogram.Conclusion: The figure is a square. Step 1 Identify the hypothesis p and the conclusion q of the true conditional. p: A figure is a square. q: The figure is a parallelogram. Example 2

  17. Law of Detachment Step 2 Analyze the conclusion. The given statement the figure is a parallelogram satisfies the conclusion q of the true conditional. However, knowing that a conditional statement and its conclusion are true does not make the hypothesis true. The figure could be a rectangle. Answer: The conclusion is invalid. Example 2

  18. The following is a true conditional. Determine whether the conclusion is valid based on the given information. Given: If a polygon is a convex quadrilateral, then the sum of the interior angles is 360. ABCD is a convex quadrilateral.Conclusion: The sum of the interior angles of ABCD is 360. A. valid B. not valid C. cannot be determined Example 2

  19. Judge Conclusions Using Venn Diagrams Determine whether the conclusion is valid based on the given information. If not, write invalid. Explain your reasoning using a Venn diagram.Given: If a triangle is equilateral, then it is an acute triangle. The triangle is equilateral.Conclusion: The triangle is acute. Analyze Draw a Venn diagram. According to the conditional, if a triangle is equilateral, then it is acute. Draw a circle for acute triangles. Draw a circle for equilateral triangles inside the circle for acute triangles. Example 3

  20. Judge Conclusions Using Venn Diagrams Formulate We know that the given triangle is equilateral. Determine Since all equilateral triangles are acute, the conclusion is valid. Answer: From the information given, all equilateral triangles are acute, so the conclusion is valid. JustifyNo matter where you draw a dot for an equilateral triangle, it will always be inside the circle for acute triangles. Therefore, the conclusion is valid. Example 3

  21. Judge Conclusions Using Venn Diagrams EvaluateVenn diagrams are a great way to visualize groups and subgroups. By representing the conditional as a Venn diagram, it becomes clear that all equilateral triangles are acute triangles. Our conclusion is valid. Example 3

  22. Determine whether the conclusion is valid based on the given information. If not, write invalid. Use a Venn diagram to help you.Given: If a figure is a square, then it has 4 right angles. A figure has 4 right angles.Conclusion: The figure is a square. A. valid B. invalid Example 3

  23. Concept

  24. Law of Syllogism Determine which statement follows logically from the given statements. (1) If Jamal finishes his homework, he will go out with his friends.(2) If Jamal goes out with his friends, he will go to the movies. AIf Jamal goes out with his friends, then he finishes his homework.BIf Jamal finishes his homework, he will go to the movies.CIf Jamal does not go to the movies, he does not go out with his friends.DThere is no valid conclusion. Example 4

  25. Law of Syllogism Read the Item Let p, q, and r represent the parts of the given conditional statements. p: Jamal finishes his homework. q: He goes out with his friends. r: He goes to the movies. Solve the Item Analyze the logic of the given conditional statements using symbols. Statement (1): p → q Statement (2): q → r Example 4

  26. Law of Syllogism Both statements are considered true. By the Law of Syllogism, p → r is also true. Write the statements for p → r. If Jamal finishes his homework, then he will go to the movies. Answer: The correct choice is B. Example 4

  27. Determine which statement follows logically from the given statements. (1) If your alarm clock goes off in the morning, then you will get out of bed.(2) If you ride a bus, then you go to work. A. If you ride a bus, then you get out of bed. B. If your alarm clock goes off, then you go to work. C. If you go to work, then you get of out bed. D. There is no valid conclusion. Example 4

  28. Apply Laws of Deductive Reasoning Draw a valid conclusion from the given statements, if possible. Then state whether your conclusion was drawn using the Law of Detachment or the Law of Syllogism. If no valid conclusion can be drawn, write no conclusion and explain your reasoning.Given: If it snows more than 5 inches, school will be closed. It snows 7 inches. p: It snows more than 5 inches. q: School will be closed. Answer: Since It snows 7 inches satisfies the hypothesis, p is true. By the Law of Detachment, a valid conclusion is School is closed. Example 5

  29. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, select invalid.(1) If a children’s movie is playing on Saturday, Janine will take her little sister Jill to the movie. (2) Janine always buys Jill popcorn at the movies.(3) If a children’s movie is playing on Saturday, Jill will get popcorn. A. Law of Detachment B. Law of Syllogism C. invalid Example 5

  30. LESSON 2–4 Deductive Reasoning

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