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1.3 – The Look and Language of Logic

Objectives

  • Explore Venn diagrams as representations of conditional statements.
  • Investigate the converse, inverse, and contrapositives of conditional statements.
  • Use the basic symbols of logic to write conditional statements in shorthand.
  • Use syllogisms to form chains of (or make connections between) ideas.

Key Terms

  • contrapositive – a statement in the form “If not B, then not A,” given the statement “If A, then B.”
  • converse – a statement in the form “If B, then A,” given the statement “If A, then B.”
  • Counterexample – an exception to a proposed general rule or law
  • inverse – a statement in the form “If not A, then not B,” given the statement “If A, then B.”
  • syllogism – a form of deductive reasoning that combines two or more related conditional statements in order to arrive at a conclusion.
  • Venn diagram – a diagram that uses two or more circles or other shapes to represent sets. Elements that belong to more than one set are placed in the areas where the circles overlap.

Notes

Rules of Logic
  • For a statement to be true, it must be true for every possible case.
  • A statement is only TRUE if there are NO exceptions.
  • For a statement to be false, it must be false in at least one possible case.
  • A statement is FALSE even if there is ONLY ONE exception.

 

Language of Logic
  • Venn diagrams can be used to represent conditional logic.  Watch the video below.

(Apex Learning, 2015)


  • Summary
    • Remember: IF statements go on the INSIDE of Venn diagrams!
Geo A 1.03 - Venn Diagram

(Apex Learning, 2015)

GeoA 01.03 - Conditional Statement Titles

(Apex Learning, 2015)

GeoA-01.03-Conditional-Statements-Animation

(Apex Learning, 2015)

 

  • Shorthand symbols

(Apex Learning, 2015)


  • Summary
GeoA-01.03 Shorthand

(Apex Learning, 2015)

  • Examples
Geo A 1.03 - If Then Symbols

(Apex Learning, 2015)

 

  • Syllogism: a chain of conditional statements.
    • In the chain, the first and last statements combine to make up the conclusion.
GeoA-01.03 Syllogism

(Apex Learning, 2015)


  • Watch the video (below) that shows how to write the conclusion of a syllogism.

  • Example
    • If I eat pie, then I will be happy.  If I am happy, then I will do my homework.  If I do my homework, then I will get my degree.  If I get my degree, then I will get a good job!  So, if I eat pie, then I will get a good job!
    • Shorthand
      • p\implies h and
      • h\implies w and
      • w\implies d and
      • d\implies j
      • So, in conclusion, p\implies j.

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