1. Geometry as a Deductive System or Structure

2. D Inductive Reasoning – observe specific cases to form a general rule Example: Little kid burns his hand five days in a row and forms a general rule:“the stove is always hot and dangerous.”

3. D Deductive Reasoning – apply a (known) general rule to a specific case Example: High school physics student applies facts that electricity creates friction and friction produces heat to know that he should not touch the stove when electricity is flowing to the burner.

4. *Our two-column proofs use Deductive Reasoning! Every specific statement in the left column is justified by a known reason in the right column!

5. *Our two-column proofs use Deductive Reasoning! 5 Categories of Reasons in a Proof Given Definitions Properties Postulates Theorems (previously proven)

6. D Conditional Statement or Implication – A mathematical statement in “if – then” format

7. D Conditional Statement or Implication - A mathematical statement in “if – then” format D Hypothesis – the if clause (becomes the given in a proof) D Conclusion – the then clause (becomes the prove statement in a proof)

8. LOGIC SYMBOLS: read aloud as “if…then” or “implies” read aloud as “if and only if” or “is equivalent to”

9. LOGIC SYMBOLS: ~ read as “not” or “the negation of” a, b, c, n, m, … We use letters as logic variables. Each represents some “phrase”

10. Examples of a conditional: a = today is Thursday b = tomorrow is Friday a b could be read as: “If today is Thursday, then tomorrow is Friday.”

11. Examples of a conditional: a = today is Thursday b = tomorrow is Friday ~a b read as “If today is NOT Thursday, then tomorrow is Friday.”

12. D Converse – of an implication swaps the hypothesis and conclusion For a  b, the CONVERSE is b  a read as: “If tomorrow is Friday, then today is Thursday.” **Here the original statement is true AND the converse is also true.

13. D Inverse – of an implication negates the hypothesis and conclusion For a  b, the INVERSE is ~a  ~b read as “If today is not Thursday, then tomorrow is not Friday.” **Here the original statement is true AND the inverse is also true.

14. D Contrapositive – of an implication is the converse and inverse at the same time For a  b, the CONTRAPOSITIVE is ~ b  ~ a read as “If tomorrow is not Friday, then today is not Thursday.” **Here the original statement is true AND the contrapositive is also true.

15. *Converse of definitions are always true so we say that definitions are reversible. *Converse of other true statements (e.g. theorems) may be true or false. Anything goes, so you must prove them!

16. *Inverse of a true statements (e.g. theorems) may be true or false. Anything goes, so you must prove them!

17. Inverse Contrapositive Contrapositive *Inverse of a true statements (e.g. theorems) may be true or false. Anything goes, so you must prove them! *Contrapositive of a true statement is always true. *Contrapositive of a false statement is always false.

18. Chain of Reasoning D Chain of Reasoning – a sequence of conditional statements that can be summarized as one implication. a  b b  c c  d d  e This is logically equivalent to _?_

19. * To form a chain of reasoning, look for the “singletons” to use as the start and finish of your chain. To make the “daisy chain” or domino effect work, you may swap a statement with its contrapositive. (But converses or inverses would be an illegal swap because they could be false.)

20. F orm a chain of reasoning and give a one statement summary: n  r g  ~r z  ~e ~n  z This is logically equivalent to _?_

21. F orm a chain of reasoning and give a one statement summary: n  r g  ~r z  ~e ~n  z g  ~r ~r  ~n ~n  z z  ~e This is logically equivalent to _?_

22. F orm a chain of reasoning and give a one statement summary: n  r g  ~r z  ~e ~n  z g  ~r ~r  ~n ~n  z z  ~e This is logically equivalent to g  ~e

23. A Venn Diagram can be used to show set and subset relationships. Often useful illustrations in logic. Ex. If Bob lives in Tosa, then he lives in WI.

24. Ex. If Bob lives in Tosa, then he lives in WI. People that live in the US People that live in Tosa People that live in WI

25. Ex. If Bob lives in Tosa, then he lives in WI. People that live in the US People that live in Tosa People that live in WI Bob B

26. Ex. Use the Venn diagram to see which quadrilaterals are rectangles and rhombuses at the same time. rectangles trapezoids rhombuses squares Quads parallelograms