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1-7: Deductive Structure

1-7: Deductive Structure . Honors Geometry. Deductive Structure. Conclusions are based on previous statements. It contains: 1) undefined terms 2) postulates (assumptions) 3) definitions 4) Theorems & other conclusions. Undefined Terms. point, line. Postulates .

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1-7: Deductive Structure

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  1. 1-7: Deductive Structure Honors Geometry

  2. Deductive Structure • Conclusions are based on previous statements. • It contains: 1) undefined terms 2) postulates (assumptions) 3) definitions 4) Theorems & other conclusions

  3. Undefined Terms • point, line

  4. Postulates • unproved assumptions (not done yet)

  5. Definitions • states the meaning of a term or idea. • are reversible -- often written in the form “If p, then q.” (conditional) • p = hypothesis; q = conclusion • Can also be written p  q • The converse of this is q  p • Look at the proof to see which to use.

  6. EXAMPLE 1: • Consider the definition: “If an angle measures 90, then it is a right angle.” • This is called a __________. • The hypothesis is __________. • The conclusion is ____________. • What is the converse? ______________.

  7. Not always reversible • Theorems & postulates are not always reversible. • EX: If two angles are right angles, then they are congruent. • Converse: If two angles are congruent, then they are right angles. (true?)

  8. EXAMPLE 3: • Let p be the statement “you are a freshman” • Let q be the statement “your ID starts with the numbers 2014” • Is the conditional (p q) true? • What is the converse (q  p)? • Is the converse true?

  9. 1-8: Statements of Logic

  10. Logic statements • The negation of p is “not p” noted ~p. • ~ ~ p = not “not p” = p. • For example, the following: • p = I am smart • ~p = I am not smart • ~ ~ p = I am smart (not not smart)

  11. Things to Know • conditional p  q • converse q  p • inverse ~p  ~q • contrapositive ~q  ~p

  12. Try this • Let p = You are in NJ • Let q = You are in the US • conditional (p  q) _______ • converse (q  p) _______ • inverse (~p  ~q) ________ • contrapositive (~q  ~p) _______

  13. A Theorem • If a conditional statement is true, then the contrapositive of the statement is also true. • (  means “is the logical equivalent of”. This is used in the biconditional -- you’ll see in HW # 3) • Thus, the theorem can be also written: p q  ~q  ~p

  14. Chain Rule • A chain of reasoning: • If p q, and q r, then p  r

  15. Venn Diagrams • Draw a Venn Diagram to represent: • If you are in NJ, then you are in the US.

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