html5-img
1 / 16

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 .

lena
Download Presentation

1-7: Deductive Structure

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related