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# 04 geom cond

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### 04 geom cond

1. 1. CONDITIONAL STATEMENTS
2. 2. CONDITIONAL STATEMENT A conditional statement is a statement in IF and THEN form. The IF part is called the hypothesis and the THEN part is called the conclusion.
3. 3. CONDITIONAL STATEMENT IF A, then B. A  B
4. 4. If you buy a lipstick in the right place, then it’s OK to buy the wrong lipstick.
5. 5. Hypothesis: You buy a lipstick in the right place. Conclusion: It is OK to buy the wrong lipstick.
6. 6. NEGATION The negation of A is “not A”. ~A means “not A”. S: It is raining today. ~S: It is not raining today.
7. 7. TRUTH VALUE Truth value of a statement is either TRUE or FALSE. (Valid vs. Invalid)
8. 8. TRUTH VALUE A: 2011 is the year of the rabbit. Truth value: TRUE B: Water is solid. Truth value: False
9. 9. TRUTH VALUE A statement and its negation have different truth value. B: A frog is a bird. (FALSE) ~B: A frog is not a bird. (TRUE)
10. 10. DERIVED STATEMENTS CONDITIONAL INVERSE A  B ~A  ~B CONVERSE CONTRAPOSITIVE B  A ~B  ~A
11. 11. THEOREM A conditional and its corresponding contrapositive are logically equivalent. (Same truth value). The converse and inverse of a conditional are logically equivalent. (Same truth value)
12. 12. BICONDITIONAL CONDITIONAL A  B (TRUE) CONVERSE B  A (TRUE) BICONDITIONAL A <--> B
13. 13. BICONDITIONAL BICONDITIONAL A <--> B A if and only if B.
14. 14. DEDUCTIVE REASONING
15. 15. If p  q is true and p is true, then q is also true. [(pq) ^ p]  q
16. 16. If p  q and q  r are true, then p  r is also true. [(pq) ^ (qr)]  (pr)
17. 17. All AA students are female. “If a student is an AA student, then the student is a female.” (TRUE) FACT/Given: Sam is an AA student. (TRUE) Conclusion: Sam is female. (TRUE) by Law of Detachment
18. 18. All AA students are female. “If a student is an AA student, then the student is a female.” (TRUE) All females have XY chromosomes. “If you are female, then you have XY chromosomes.” Conclusion: If a student is an AA student, then the student has XY chromosomes.. (TRUE) by Law of Syllogism AA  F and F  XY therefore AA  XY
19. 19. PROVING
20. 20. To prove a conjecture, we apply deductive reasoning. To prove something we need to supply a proof. Truth is based on solid evidences (proofs). A proof is a logical argument in which each statement you make is supported by a statement that is accepted as true Forms of Proof in Geometry INFORMAL – essay form of a proof; spontaneous and descriptive/narrative FORMAL – organized and well-structured
21. 21. A group of algebraic steps used to solve problems form a deductive argument.
22. 22. A two-column proof, or formal proof, contains statements and reasons organized in two columns.