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Understanding Logic and Advanced Geometry Concepts

Explore topics such as statements, truth values, conjunctions, disjunctions, and truth tables in advanced geometry. Learn how to evaluate compound statements and their truth values. Practice problem-solving to enhance your understanding.

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Understanding Logic and Advanced Geometry Concepts

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  1. Logic Eric Hoffman Advanced Geometry PLHS Sept. 2007

  2. Key Topics • Statement – any sentence that is either true of false, but not both • Often represented by using a letter such as p or q • Truth Value – the truth of falsity of a statement • Negation – the negation of a statement has opposite meaning as well as opposite truth value • Ex. : p: Perry is a town in Kansas (~p) not p: Perry is not a town in Kansas

  3. Key Topics • Compound Statement – when two or more statements are joined by using the words “and” or “or” • Conjunction – compound statement formed when using the word “and” to join two or more statements • p Λ q read: p and q • Disjunction – compound statement formed when using the word “or” to join two or more statements • p V q read: p or q

  4. Key Topics • Conjunction Truth Values : for a conjunction to be true, both p and q have to be true. • Ex. : Perry is a town in Kansas and Perry is a smaller town than Topeka. p : Perry is a town in Kansas q : Perry is a smaller town than Topeka Both true so p Λ q is true.

  5. Key Topics • Conjunction Truth Values : p : Perry has a population of 50,000 people. q : Perry is a town in Kansas. q: is true, but p: is not true, so p Λ q is not true.

  6. Key Topics • Disjunction Truth Values – for a disjunction to be true either p has to be true, q has to be true, or both are true p: Triangles have three sides q: Equilateral triangles have 3 congruent sides r: Scalene triangles have 2 congruent sides What are the truth values for the following: p V q , p V r , q V r , p Λ q , q Λ r , p Λ r

  7. Key Topics • Truth Values – the negation of a statement always has the opposite truth value p: Triangles have three sides True ~ p: Triangles do not have three sides False q: Equilateral triangles have 3 congruent sides r: Scalene triangles have 2 congruent sides What are the truth values for the following: p V ~ q , ~ p V r , ~ q V r , p Λ ~ q , ~ q Λ r , p Λ ~ r

  8. Key Topics • Truth Tables – convenient way to organize truth values.

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  11. Key Topics • In Class Example : p : A rectangle has 4 congruent angles q : A rectangle is a square r : A square is a rectangle Construct the truth table for: (q V ~r) Λ (p Λ ~q)

  12. Key Topics • Homework: Pgs. 73, 74 18-44 even, 56-72 even 23 problems!!!

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