What is Geometry

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What is Geometry?

• The pure mathematics of points and lines and
  curves and surfaces .
• Algebra Geometry
• Definitions Definitions, Non-definitions
• absolute value midpoint/ point
• Formulas Postulates, Theorems
• difference of two squares Two distinct points
  determine a line.
• Methods Proofs
• FOIL Constructions

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Section 1.1

• Statements Reasoning

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Statement

• Group of words symbols, classified collectively
  as true or false, simple or compound. Questions,
  commands are not statements!
• Simple Snow is cold.
• Compound made up of two or more simple
  statements, connected by and, or, ifthen,
  not.

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Conjunction

• Compound statement
• Signified by and denoted by P and Q
• True only if both are true.
• False if either one or both are false
• Ex. P Snow is cold Q rain is wet
• P and Q Snow is cold and rain is wet
• Ex P It is hot Q it is snowing
• P and Q It is hot and it is snowing

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Disjunction

• Compound statement
• Signified by or denoted by P or Q
• Ex P You can have ice cream for dessert
• Q You can have strawberries for dessert
• P and Q
• You can have ice cream or strawberries for
  dessert.
• False only if both are false.
• True if either one or both are true

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Negation

• Not
• Changes the truth value of the statement to its
  opposite.
• Let P be the statement My car is white.
• Then the negation of P is not P or P.
• My car is not white.
• What is the negation of Some?
• Ex Some men have beards
• Not one man has a beard.

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Conditional statement

• Also called implication
• Ifthen, symbolized by using ?
• Ex. P The sun is shining
• Q I can see my shadow,
• P ?Q If the sun is shining, then I can see my
  shadow.
• P is the hypothesis and Q is the conclusion.

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3 types of Reasoning1. Intuition

• Intuition An idea leading to a statement of a
  theory.
• You enter the bank and the line is very long.
  You conclude that you will have a long wait.
• Before the opening kickoff of the first game of
  the season, Bill predicts his team will win.

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2. Inductive reasoning

• Induction Using specific observations to draw a
  general conclusion (from specific to general).
• You find a bag of tennis balls and the first 3
  are flat. You conclude that the whole bag is
  flat.
• After examining and diagnosing several patients,
  the doctor concludes that there is a flu epidemic
  in that area.
• Involves examining a few examples, observing a
  pattern, and then assuming that the pattern will
  never end.
• Not a valid proof, although it often suggests
  statements that can be proved by other methods.

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Example of Inductive Reasoning

• String of odd integers Sum
• 1 3 4
• 1 3 5 9
• 1 3 5 7 16
• 1 3 5 7 9 25
• Do you notice a pattern?
• What conclusion can you draw?

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3. Deductive reasoning

• Deduction Accept that certain assumptions are
  true which guarantee a specific conclusion.
• You know that the movie is two hours and starts
  at 8 pm.
• You conclude that it will end at 10 pm.
• If an integer is even, then it is divisible by
  two. Since 14 is an even integer, it is
  divisible by two.
• May be considered the opposite of inductive
  reasoning.
• Uses accepted facts, i.e. undefined terms,
  defined terms, postulates, previously
  established theorems, to reason in a step-by-step
  fashion until a desired conclusion is reached.

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An Example of Deduction

• Assume the following 2 postulates are true
• All last names that have 7 letters with no vowels
  are the names of Martians.
• All Martians are 3 feet tall.
• Prove that Mr. Xhzftlr is 3 feet tall.

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Proof

• Use the 2-column format
• Statements Reasons
• 1. The name is Mr. Xhzftlr. 1. Given.
• 2. Mr. Xhzftlr is a Martian. 2. All
  last names with no vowels are the
  names of Martians (Post. 1)
• 3. Mr. Xhzftlr is 3 feet tall. 3. All Martians
  are 3 feet tall (Post. 2).
• Notice that each statement has a corresponding
  justification.

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Law of DetachmentA form of deductive
reasoningIf statements 1 and 2 (premises) are
true, then the conclusion is true.

• 1. If P, then Q premises
• 2. P .
  
• ? Q conclusion
• P It is raining
• Q The field is wet
• If P ?Q
• If it is raining, then the field will be wet.
• CANNOT MAKE ANY STATEMENT CONCERNING
• IF Q THEN P. (Fallacy of the Converse)
• If it is raining then the field will be wet
• If the field is wet, does that mean it is
  raining?
• Give other options