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Inductive and Deductive Reasoning

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magee-sandoval
20 diapositives · 373 Views · Oct 25, 2014

This resource provides a comprehensive overview of inductive and deductive reasoning, including key definitions like conjecture, counterexample, conditionals, and biconditional statements. It explains the structure of conditional statements, their converses, inverses, and contrapositives, as well as the laws of detachment and syllogism used in deductive reasoning. By illustrating these concepts with examples, it clarifies how to apply these reasoning methods to validate mathematical statements and derive conclusions from premises. Ideal for students learning logic and proof strategies in mathematics.

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Inductive and Deductive Reasoning
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  1. Inductive and Deductive Reasoning

  2. Notecard 30 Definition: Conjecture: an unproven statement that is based on observations or given information.

  3. Notecard 31 Definition: Counterexample: a specific case for which a conjecture is false.

  4. Counterexample Find a counter example to show that the following conjecture is false. The sum of two numbers is always greater than the larger number.

  5. Notecard 32 Definitions: Conditionals, Hypothesis, & Conclusions: A conditional statement is a logical statement that has two parts: If ____ then _____. The hypothesis is the “if” part and it tells you what you are talking about. The conclusion is the “then” part and it describes the hypothesis.

  6. Writing a conditional statement • Writing the following statements as conditionals. Two angles that make a linear pair are supplementary. All 90o angles are right angles.

  7. Notecard 33 • The negation of a statement is the opposite of the original.

  8. Negation • Negate the following statements. The ball is red. The cat is not black.

  9. Notecard 34 Definitions: Inverse, Converse, Contrapositive • The converse of a conditional statement switches the hypothesis and conclusion. • The inverse of a conditional statement negates both the hypothesis and conclusion • The contrapositive of a conditional statement takes the inverse of the converse. (it switches and negates)

  10. Writing statements • Write the converse, inverse and contrapositive of the conditional statement: “If two angles form a linear pair, then they are supplementary.” Which of these statements are true?

  11. Notecard 35 Definition: Biconditional: If a conditional statement and its converse are both true, then we can write it as a biconditional statement by using the phrase if and only if instead of putting it in if-then form. __________ if and only if ___________. (hypothesis) (conclusion)

  12. Biconditional Statement • Write the following conditional statement as a biconditional statement. • If two lines intersect to form a right angle, then they are perpendicular.

  13. Notecard 36 The Law of Detachment This applies when one statement is conditional and a second statement confirms the hypothesis of the conditional. The conclusion is then confirmed. Here is an example.

  14. Deductive Reasoning If it is Friday, then Mary goes to the movies. It is Friday. What conjecture can you make from the above statements?

  15. Deductive Reasoning • If two angles form a linear pair, then they are supplementary. • Angle 1 and Angle 2 are a linear pair.

  16. Deductive Reasoning • If two angles form a linear pair, then they are supplementary. • Angle 1 and Angle 2 are supplementary.

  17. Notecard 37 The Law of Syllogism • This applies when you have two conditional statements. The conclusion of one, confirms the hypothesis of the other. In this case our result is still a conditional with the first hypothesis and the second conclusion. (I call this the “Oreo Cookie” Law.) Here is how it works…

  18. Deductive Reasoning If it is Friday, then Mary goes to the movies. If Mary goes to the movies then she gets popcorn. • Combine the two above conditional statements into one conditional statement.

  19. Deductive Reasoning • If two angles form a linear pair, then they are supplementary. • If two angles are supplementary then their sum is 180 degrees.

  20. Deductive Reasoning • If a polygon is regular, then all angles in the interior of the polygon are congruent. If a polygon is regular, then all of its sides are congruent. • Why can’t these two statements be combined like the last example.