WHY DO WE NEED TO STUDY GEOMETRY?

2. WHY DO WE NEED TO STUDY GEOMETRY?

3. WE HAVE TO STUDY GEOMETRY TO: TO PREPARE US FOR HIGHER MATHEMATICS.

4. WE HAVE TO STUDY GEOMETRY TO: UNDERSTAND AND APPRECIATE OUR NATURAL AND MAN-MADE ENVIRONMENT.

5. WE HAVE TO STUDY GEOMETRY TO: PROVIDE US WITH MANY IMPORTANT FACTS OF PRACTICAL VALUE.

6. WE HAVE TO STUDY GEOMETRY TO: ENHANCE OUR ANALYTICAL SKILLS TO ENABLE US TO EXPRESS OUR THOUGHTS ACCURATELY AND TRAIN US TO REASON LOGICALLY.

7. What You Need To Know

8. Conditional -it is something which states that one statement implies another. A conditional contains two parts: the condition and the conclusion, where the former implies the latter. Before one can start to understand logic, and thereby begin to prove geometric theorems, one must first know a few vocabulary words and symbols.

9. CONDITIONAL A conditional is always in the form "If statement 1, then statement 2.“ Statement 1 – hypothesis Statement 2 – conclusion

10. CONDITIONAL which is read as "If p, then q" where p and q are statements. In most mathematical notation, a conditional is often written in the form p ⇒ q,

11. CONDITIONAL Examples If two angles have equal measures, then they are congruent. If two segments are congruent, then they have equal measures.

12. But these conditionals can be written in the if-then form. However, there are times when conditionals do not appear in the if-then form.

13. CONDITIONAL Example All right angles are congruent. If all angles are right, then they are congruent. FOR MORE EXAMPLES: SEE PAGE 59 ,GEOMETRY TEXTBOOK

14. Conditional statesments are not always written with the ‘if’ clause first. Example Example If x² = 4, then x = 2 x² = 4 implies x = 2 x² = 4only if x = 2 x = 2 if x² = 4 General Form If p, then q p implies q p only if q q if p

15. CONVERSE The converse of a logical statement is when the conclusion becomes the condition and vice versa; i.e., p ⇒ q becomes q ⇒ p.

16. CONDITIONAL Example If two angles have equal measures, then they are congruent. CONVERSE If two angles are congruent, then they have equal measures. FOR MORE EXAMPLES: SEE PAGE 59 ,GEOMETRY TEXTBOOK

17. CONVERSE However, the converse of a conditional is not always true.

18. CONDITIONAL Example "If someone is a woman, then they are a human" CONVERSE "If someone is a human, then they are a woman." The converse is FALSE because a man is also a human.

19. CONVERSE Switching the hypothesis and the conclusion forms a converse of a conditional. i.e., “if p, thenq” becomes “if q, thenp”.

20. NOT If a statement is preceded by "NOT," then it is evaluating the opposite truth value of that statement. The symbol for "NOT" is For example, if the statement p is "Elvis is dead," then ¬p would be "Elvis is not dead."

21. CONDITIONAL Examples Then –r would be if r is "¬". "All men have hair," "All men do not have hair“ "No men have hair."

22. NOT Do not confuse this with "Not all men have hair" or "Some men have hair." The "NOT" should apply to the verb in the statement: in this case, "have." ¬p can also be written as NOT p or~p. NOT p may also be referred to as the "negation of p.“

23. CONDITIONAL Examples Negation ( p) 2. Sam is sleeping in class. “ It is not true that Sam is sleeping in class”. “Sam is not sleeping in class."

24. INVERSE The inverse of a conditional says that the negation of the condition implies the negation of the conclusion.

25. CONDITIONAL Example p ⇒ q read as “If p, then q”. INVERSE -p ⇒ -q read as “If NOT p, then NOT q”. Like a converse, an inverse does not necessarily have the same truth value as the original conditional.

26. CONTRAPOSITIVE An inverse but switched around with the p and q. For example, Statement: If p, then q Inverse: If not p, then not q Contrapositive: if not q, then not p The statement is always true with the contrapositive, but a statement is not logically equivalent to its converse or to its inverse.

27. CONDITIONAL Example INVERSE If it is not raining then the ground is not getting wet. If it is raining then the ground is getting wet. CONTRAPOSITIVE If the ground is not getting wet then it is not raining.

28. CONDITIONAL Example INVERSE If the cat will NOT run then the dog will NOT chase the cat. CONVERSE If the cat will run then the dog will chase the cat. If the dog will chase the cat then the cat will run. CONTRAPOSITIVE If the dog will NOT chase the cat then the cat will NOT run.

29. CONDITIONAL INVERSE “If negation of p then negation of q” CONVERSE “If p then q” “If q then p” CONTRAPOSITIVE “If negation of q then negation of p” NOTE: The conditional statement and its contra positive are logically equivalent

30. BICONDITIONAL A biconditional is conditional where the condition and the conclusion imply one another. A biconditional starts with the words "if and only if." For example, "If and only if p, then q" means both that p implies q and that q implies p.

31. SYMBOLS Iff: is a shortened form of "if and only if." It is read as "if and only if.“ ⇔: The symbol which denotes a biconditonal. p ⇔ q is read as "If and only if p, then q.“

32. PREMISE A premise is a statement whose truth value is known initially. For example, if one were to say "If today is Thursday, then the cafeteria will serve burritos," and one knew that what day it was, then the premise would be "Today is Thursday" or "Today is not Thursday."

33. CONJUNCTION -is compound statement composed of two simple statements joined by the word “and”. The symbol ∧ is used to represent the word “and”.

34. DISJUNCTION is compound statement composed of two simple statements joined by the word “or ”. The symbol∨is used to represent the word “or”.

35. Simple statements Example p Mom plays the guitar. q Dad plays the piano. CONJUNCTION p ∧ q " Mom plays the guitar and Dad plays the piano ."

36. Simple statements Example p Mom plays the guitar. q Dad plays the piano. DISJUNCTION p Vq " Mom plays the guitar or Dad plays the piano ."

37. THEREFORE ∴: The symbol for "therefore." p ∴ q means that one knows that p is true (p is true is the premise), and has logically concluded that q must also be true.

38. Example Therefore, It is a trapezoid. It is a square or it is a trapezoid. It is not a square.

39. DEFINITION conjecture: An unproven statement that is based on observations.

40. How much do you know Get one-half crosswise Given the conditional statement, state the inverse, converse and its contra positive.

41. CONDITIONAL INVERSE CONVERSE “If today is Tuesday then tomorrow is Wednesday” CONTRAPOSITIVE

42. Suppose p stands for “Hawks swoop” and q stands for “Gulls glide”. Express the following in symbolic form each of the following statements. • 1. Hawks swoop or gulls glide. • 2. Gulls do not glide. • 3. Hawks do not swoop or gulls do not glide. • 4. Hawks do not swoop and gulls do not glide.

43. Let’s see if your answers are correct.

44. CONDITIONAL INVERSE “If today is not Tuesday then tomorrow is not Wednesday” “If today is Tuesday then tomorrow is Wednesday” CONVERSE “If tomorrow is Wednesday then today is Tuesday” CONTRAPOSITIVE “If tomorrow is not Wednesday then today is not Tuesday”

45. Suppose p stands for “Hawks swoop” and q stands for “Gulls glide”. Express the following in symbolic form each of the following statements. • 1. Hawks swoop or gulls glide. • 2. Gulls do not glide. • 3. Hawks do not swoop or gulls do not glide. • 4. Hawks do not swoop and gulls do not glide. p V q  q  p V q  p Ʌ  q

46. Perfect score is 7. Pass your paper.

47. TYPES OF REASONING Inductive Versus Deductive Reasoning