Logical Arguments in Mathematics

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.

Deductive ProofStep by step process of drawing conclusions

based on previously known truths.

Properties of Deductive Proofs: Uses “Top Down” Logic and ReasoningTakes 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)

Deductive Reasoning: Flow Chart

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 In logical analysis, variables no longer represent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 Franceq – The moon is made of green cheese

What does the conditional statement: If p, then q say?

Conditional StatementsDeductive 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:

Using Conditional Statements to Complete Deductive Proofs

Look for the assumption of the hypothesisFollow 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?

Problems with Deductive ArgumentsErrors 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.

Practice with Deductive Arguments1. 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

Logical Arguments in Mathematics and Real Life

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.

Inductive ProofThe process of arriving at a conclusion based

in a set of observations.

Properties of Inductive Proofs: Uses “Bottom Up” Logic and ReasoningHighly based on patternsTakes specific incidents of an event to develop

an overall conclusionDownfall… NOT an acceptable form of proof

Inductive Reasoning: Flow Chart

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.

Benefits to Inductive ArgumentsA 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

Examples of InductionNumerical Patterns: Find the next two terms of each

sequence1, 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.

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.

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?