Obj. 10 Deductive Reasoning

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The student is able to (I can):
Apply the Law of Detachment and the Law of Syllogism in logical reasoning
Write and analyze biconditional statements.

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Obj. 10 Deductive Reasoning

  1. 1. Obj. 10 Deductive Reasoning Objectives The student is able to (I can): • Apply the Law of Detachment and the Law of Syllogism in logical reasoning • Write and analyze biconditional statements.
  2. 2. Recall from Inductive Reasoning: • One counterexample is enough to disprove a conjecture. • If we can’t come up with a counterexample, how can we prove that a conjecture is true for every case?
  3. 3. deductive reasoning The process of using logic to draw conclusions from given facts, definitions, and properties. Inductive reasoning uses specific cases and observations to form conclusions about general ones (circumstantial evidence). Deductive reasoning uses facts about general cases to form conclusions about specific cases (direct evidence).
  4. 4. Example Decide whether each conclusion uses inductive or deductive reasoning. 1. Police arrest a person for robbery when they find him in possession of stolen merchandise. Inductive reasoningInductive reasoningInductive reasoningInductive reasoning 2. Gunpowder residue tests show that a suspect had fired a gun recently. Deductive reasoningDeductive reasoningDeductive reasoningDeductive reasoning
  5. 5. Most of our conjectures can be phrased as “if p then q.” This is often written p → q. Law of Detachment • If p → q is a true statement and p is true, then q is true.
  6. 6. Examples Determine if the conjecture is valid by the Law of Detachment. Given: If a student passes his classes, the student is eligible to play sports. Ramon passed his classes. Conjecture: Ramon is eligible to play sports. Given: If you are tardy 3 times, you must go to detention. Shea is in detention. Conjecture: Shea was tardy at least 3 times. validvalidvalidvalid not validnot validnot validnot valid
  7. 7. Examples Law of Syllogism • If p → q and q → r are true statements, then p → r is a true statement. Determine if each conjecture is valid by the Law of Syllogism. Given: If a number is divisible by 4, then it is divisible by 2. If a number is even, then it is divisible by 2. Conjecture: If a number is divisible by 4, then it is even. x: A number is divisible by 4 y: A number is divisible by 2 z: A number is even x → y and z → y; therefore, x → z not validnot validnot validnot valid
  8. 8. Determine if each conjecture is valid by the Law of Syllogism. Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal. Conjecture: If an animal is a dog, then it has hair. x: An animal is a mammal y: It has hair z: An animal is a dog x → y and z → x, therefore z → y or z → x and, x → y therefore z → y validvalidvalidvalid
  9. 9. biconditional statement A statement whose conditional and converse are both true. It is written as “p if and only if qp if and only if qp if and only if qp if and only if q”, “p iff qp iff qp iff qp iff q”, or “pppp ↔↔↔↔ qqqq”. This means that p → q is true, and q → p is true.
  10. 10. To write the conditional statement and converse within the biconditional, first identify the hypothesis and conclusion, then write p → q and q → p. Example: Two lines are parallel if and only if they never intersect. Conditional: If two lines are parallel, then they never intersect. Converse: If two lines never intersect, then they are parallel.
  11. 11. Example Write the conditional and converse from the biconditional statement. A solution is a base iff it has a pH greater than 7. Conditional: If a solution is a base, then it has a pH greater than 7. Converse: If a solution has a pH greater than 7, then it is a base.
  12. 12. Example Writing a biconditional statement: 1. Identify the hypothesis and conclusion. 2. Write the hypothesis, “if and only if”, and the conclusion. Write the converse and biconditional from: If 4x + 3 = 11, then x = 2. Converse: If x = 2, then 4x + 3 = 11. Biconditional: 4x + 3 = 11 iff x = 2.
  13. 13. Remember, for a biconditional to be true, both the conditional and the converse must be true. Determine if the biconditional is true, or if false, give a counterexample. A quadrilateral is a square if and only if it has four right angles. Conditional: If a quadrilateral is a square, then it has four right angles. TRUETRUETRUETRUE Converse: If a quadrilateral has four right angles, then it is a square. FALSEFALSEFALSEFALSE (it could be a rectangle)
  14. 14. Any definition in geometry can be written as a biconditional. Write each definition as a biconditional: 1. A rectangle is a quadrilateral with four right angles. • A quadrilateral is a rectangle iff it has four right angles. 2. Congruent angles are angles that have the same measure. • Angles are congruent angles iff they have the same measure.

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