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Reasoning

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Inductive Reasoning

  • When you make a conclusion based on a pattern of examples or past events

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Conjecture

  • The conclusion you make using inductive reasoning
    • These are only educated guesses

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Counterexamples

  • The example used to prove a conjecture false
    • Ex: All dogs have spots
      • Can you find a counterexample for this?

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Find counterexamples for the following:

  • Any three points make a plane
    • Collinear points
  • Dividing any number by 2 cut’s it’s value in half
    • Zero
  • All prime numbers are odd
    • Two
  • All months have at least 30 days
    • February

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Conditional Statements

  • Statements can be written in an if-then form
    • Ex: April has 30 days.
      • If it is April, then there are 30 days in the month.

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  • Words are added or taken away to make the statement make sense.

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Hypothesis

  • The part between the “if” and “then”
    • If it is April, then there are 30 days in the month.
      • Hypothesis: It is April.
  • **You do not include the “If” or “then” when stating the hypothesis

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Conclusion

  • The part following the “then”
    • If it is April, then there are 30 days in the month.
      • Conclusion: There are 30 days in the month.
  • **You do not include the “then” when stating the conclusion

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Examples

  • Name the hypothesis and the conclusion:
    • If the year is 1777, then there are 13 stars on the U.S. flag.
      • Hypothesis: The year is 1777.
      • Conclusion: There are 13 stars on the U.S. flag.
    • If you have Ms. Holly for math, then you love math class.
      • Hypothesis: You have Ms. Holly for math.
      • Conclusion: You love math class.

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Inverse

  • Negating the hypothesis and the conclusion.
    • Ex: If it is raining, then it is cloudy.
      • Hypothesis: It is raining.
      • Conclusion: It is cloudy.
    • Inverse:
      • If it is not raining, then it is not cloudy.

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Examples:

  • Write the inverse for the following statements:
    • If the figure is a triangle, then it has 3 angles.
      • Inverse: If the figure is not a triangle, then it does not have 3 angles.
    • If you did not like algebra, then you will like geometry.
      • Inverse: If you did like algebra, then you will not like geometry.

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Converse

  • Exchanging the hypothesis and the conclusion.
    • Ex: If it is April, then there are 30 days in the month.
      • Hypothesis: It is April.
      • Conclusion: There are 30 days in the month.
    • Converse:
      • If there are 30 days in the month, then it is April.

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Examples

  • Write the converse for the following statements:
    • If you have math with Ms. Holly, then you have math with Mrs. Reisman.
      • Converse: If you have math with Mrs. Reisman, then you have math with Ms. Holly.
    • If you are in Intro to Geometry, then your class color is Orange.
      • Converse: If your class color is orange, then you are in Intro to Geometry.

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Contrapositive

  • Negating the hypothesis and conclusion of the converse.
      • **This means you are switching the hypothesis and conclusion, as well as negating them.
    • Ex: If it is raining, then it is cloudy.
      • Hypothesis: It is raining.
      • Conclusion: It is cloudy.
    • Contrapositive
      • If it is not cloudy, then it is not raining.

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Examples

  • Write the contrapositive for the following statements
    • If you have an iphone, then you have AT&T
      • Contrapositive: If you do not have AT&T, then you do not have an iphone.
    • If you are allergic to dogs, then you have a cat.
      • Contrapositive: If you do not have a cat, then you are not allergic to dogs.

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Putting it all together

  • All collinear points lie on the same line.
  • Conditional:
    • If the points are collinear, then they lie on the same line
  • Hypothesis:
    • The points are collinear
  • Conclusion:
    • They lie on the same line

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Putting it all together

  • Conditional:
    • If the points are collinear, then they lie on the same line.
  • Inverse:
    • If the points are not collinear, then they do not lie on the same line.
  • Converse:
    • If they lie on the same line, then the points are collinear.
  • Contrapositive:
    • If they do not lie on the same line, then the points are not collinear.

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Deductive Reasoning

  • Reasoning based on facts, rules, or definitions
    • Ex: All even numbers are divisible by two. Eight is an even number; therefore it is divisible by two.

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Inductive vs. Deductive

  • Inductive is something you notice
  • Deductive is something you know

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Inductive or deductive?

  • Is the following inductive or deductive reasoning?
    • I've noticed previously that every time I kick a ball up, it comes back down, so I guess this next time when I kick it up, it will come back down, too.
      • Inductive
    • Newton’s law states: Everything that goes up must come down. And so, if you kick the ball up, it must come down.
      • Deductive

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Can deductive reasoning be false?

  • Let’s find out