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Logical Arguments in Mathematics

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A proof is a collection of statements and reasons in a logical order used to verify universal truths. However… depending upon the type of proof the definition can and will change.

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Deductive Proof Step by step process of drawing conclusions based on previously known truths. Properties of Deductive Proofs: Uses “Top Down” Logic and Reasoning Takes a general statement made about an entire class of things and then applies the rule to one specific example. Only acceptable form of a proof (Scientific and mathematical)

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Deductive Reasoning: Flow Chart Postulates Theorems Definitions Theory Assumptions Testing Hypothesis Specific Case Proof Observation Conclusion Specific Case Confirmation

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Definitions used in Deductive Arguments Logical Statements: Statements that can be true or false. In logical analysis, variables no longer represent numbers… instead they represent logical statements. Most logical statements are written as conditional statements. Example: p – Paris is the capital of France q – The moon is made of green cheese What does the conditional statement: If p, then q say?

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Conditional Statements Deductive Arguments are based on conditional statements. All the postulates and theorems we are studying are conditional statements. When proving a theorem… we assume the hypothesis and show how to get the conclusion. For a conditional statement to be true consider the following:

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Using Conditional Statements to Complete Deductive Proofs Look for the assumption of the hypothesis Follow each piece of the argument carefully. Remember… very similar to the transitive property! Example: If Lyn is taller than Mark, then Mark is taller than Eddie. Lyn is taller than Mark. What can you conclude about Mark?

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Problems with Deductive Arguments Errors in deductive arguments are called fallacies. Examine the following argument. Why might it not be a “good” argument? Premise: All good basketball players are over 6 feet tall. Grant is 6 foot 3 inches tall. Conclusion: Grant is a good basketball player.

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Practice with Deductive Arguments 1. When the sun shines, the grass grows. When the grass grows, it needs to be cut. The sun is shining. What can you deduce about the grass? 2. Jim is a good barber. Everybody who gets a haircut by Jim gets a good haircut. Austin has a good haircut. What can you deduce about Austin? 3. Why is the following example of deductive reasoning faulty? Premise: Khaki pants are comfortable Comfortable pants are expensive Adrian’s pants are not khaki pants Conclusion: Adrian’s pants are not expensive

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Logical Arguments in Mathematics and Real Life

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Examine the following argument. Explain how this argument is different. Is the conclusion of the argument true? Argument 1: After picking roses for the first time, Jamie began to sneeze. She also began sneezing the next four times she was near roses. Based on these past experiences, Jamie decides that she is allergic to roses.

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Inductive Proof The process of arriving at a conclusion based in a set of observations. Properties of Inductive Proofs: Uses “Bottom Up” Logic and Reasoning Highly based on patterns Takes specific incidents of an event to develop an overall conclusion Downfall… NOT an acceptable form of proof

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Inductive Reasoning: Flow Chart

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Major Problems with Inductive Arguments Since many inductive arguments are based on patterns, there is NO guarantee that the conditions will always be true. Example: The number pi… originally it was thought that pi had an exact value, i.e recognizable pattern.

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Benefits to Inductive Arguments A hypothesis based on inductive reasoning can lead to a more careful study of a situation. Allows for more in-depth development of hypotheses for experiments. Many times theories in science, mathematics, and education are developed and tested using inductive arguments

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Examples of Induction Numerical Patterns: Find the next two terms of each sequence 1, 4, 16, 64, …, How? 18, 15, 12, 9, …, How? 10, 12, 16, 22, …, How? 8, -4, 2, -1, ½,…, How? 2, 20, 10, 100, 50…, How? Extra Credit: Write the equations to represent each of the sequences above.

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Inductive or Deductive? Examine the following scenarios. Determine if the arguments use deductive or inductive reasoning. Argument 1: Jake noticed that spaghetti has been on the school menu for the past five Wednesdays. Jake decides that the school always serves spaghetti on Wednesday. Argument 2: By using the definitions of equilateral triangles and of perimeter, Katie concludes that the perimeter of every equilateral triangle is three times the length of a side.

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Inductive or Deductive? Argument 3: Brendan observes that (-1) 2 = +1; (-1) 4 = +1; and (-1) 6 = +1. He concludes that every even power of (-1) equals +1 Argument 4: There are three sisters. Two of them are athletes and two of them like ice cream. Can you be sure that both of the athletes like ice cream. Do you reason deductively or inductively to conclude the following: At least one of the athletic sisters like ice cream?

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