University:
High School
Course:
High School Mathematics
Academic year:

2022
Views:

293

Pages:

3
Author:

Cierra S

Inductive & Deductive Reasoning: Inductive Reasoning: Using an "observed" pattern to make a general statement. Example: "The cafe serves pizza the last 5 Fridays, so you guess Friday pizza." Deductive Reasoning: Using "facts," "definitions," "laws," or "rules" to make a general statement. Laws of Logic: Law of Detachment: When given a true conditional statement (p → q) and we know p is true, then we can conclude that q is true. Law of Syllogism: Combining two conditional statements (p → q and q → r) into one. If p → q and q → r, then p → r. Additional Notes: The term "conjecture" is defined as an unproven statement based on observations. A "counterexample" is one example that proves a conjecture false. The Law of Syllogism is described as a way to "cut out the middle" of a logical argument. Extra Practice: 1. 2, 19, 17, 14, 10, ... Pattern: Subtracting consecutive odd numbers. Next two numbers: 6, 1. 2. -1, 2, -3, 5, -7, ... Pattern: Alternating signs and increasing prime numbers. Next two numbers: 11, -13. 3. C, E, G, I, K, ... Pattern: Skipping a letter. Next two letters: M, O. 4. A sequence of boxes with increasing number of lines. Pattern: Adding one more line to each box. Next two: Two boxes with 6 and 7 lines, respectively. 5. Conjecture: The sum of two negative integers is always negative. Truth Table: Several examples are given to test the conjecture. Conclusion: The conjecture is true. 6. Conjecture: The product of three consecutive nonzero integers is always even. Truth Table: Several examples are given to test the conjecture. Conclusion: The conjecture is true. 7. Counterexample for the conjecture "If n is a rational number, then n² is always less than n." Example: n = 3/2, n² = 9/4 > 3/2 8. Counterexample for the conjecture "If line k intersects plane P at point Q, then plane P is perpendicular to line k." A diagram shows a line intersecting a plane at a non-right angle. Extra Practice: In Exercises 9 and 10, use the Law of Detachment to determine what you can conclude from the given information, if possible. 9. If a triangle has equal side lengths, then each interior angle measure is 60 degrees. Given: Triangle ABC has equal side lengths. Conclusion: Each interior angle of triangle ABC is 60 degrees. 10. If a quadrilateral is a rhombus, then it has two pairs of opposite sides that are parallel. Given: Quadrilateral PQRS has two pairs of opposite sides that are parallel. Conclusion: No definite conclusion can be made as the given information doesn't match the "if" part of the conditional statement. In Exercises 11 and 12, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible. 11. If it does not rain, then I will walk to school. If I walk to school, then I will wear my walking shoes. New statement: If it does not rain, then I will wear my walking shoes. 12. If x > 1, then 3x > 3. If 3x > 3, then (3x)^2 > 9. New statement: If x > 1, then (3x)^2 > 9.