1. “Education is the most powerful weapon which you can use to change the world.” ― Nelson Mandela Do NowNelson Mandela  Put your homework assignment (examples of inductive and deductive reasoning you heard this weekend) on your desk ready to be stamped and collected.  Explain in your own words what deductive reasoning is.

2. Exit Slip Error Analysis  Compare your exit slip to this one.

3. Problem Solving at an early age

4. You be the judge Deductive Reasoning II Judging the validity of conditional statements

5. Today’s Objectives  Explain the laws used in the deductive reasoning process.  Use deductive reasoning to lead to accurate conclusions.  Use the Law of Detachment  Use the Law of Syllogism  Use Problem Solving Skills

6. Rewrite in conditional (“if-then”) form  All quadrilaterals have four sides.  If it’s a quadrilateral, then it has four sides.  Inverse?  If it’s not a quadrilateral, then it does not have four sides.  In other words, it has more or less than four sides.

7. Rewrite in conditional (“if-then”) form  A triangle has, at most, one right angle.  If it’s a triangle, then it has, at most, one right angle.  Inverse?  If it’s not a triangle, then it has more than one right angle.

8. Rewrite in conditional (“if-then”) form  Two lines in a plane always intersect at exactly one point  If there are two lines in a plane, they intersect at exactly one point  Negation?  If there are two lines in a plane, they do not intersect at exactly one point.  In other words, they do not intersect at all or they intersect at more than one point.  Counterexample: Parallel lines!

9. Some terms  Axiom – a self-evident truth that requires no proof; a statement accepted as fact  Postulate – a proposition that requires no proof  Theorem – a proposition that can be deduced from the premises or assumptions of a system  Corollary – a proposition that is incidentally proved in proving another proposition

10. Equivalence Properties  Reflexive Property  Symmetric Property  Transitive Property

11. Reflexive Property  A=A  A quantity is equal to itself  In logic, A A.  Always true in logic  If you’re a student at Simon Tech, then you’re a student at Simon Tech.  If a pentagon has five sides, then a pentagon has five sides.

12. Symmetric Property  If A=B then B=A  Always true of numbers (if x=5 then 5=x)  In logic, If A B, then B A.  Not always true.  If I eat too much I get sick. If I get sick then I eat too much.  But when is it true?  When the Biconditional statement is true.  For example, “Two lines intersect iff they are not parallel”

13. Transitive Property  Also known as the Law of Syllogism  If A=B and B=C then A=C  In logic, If AB and B C, then A C.  For example:  If the electric power is cut, then the refrigerator does not work.  If the refrigerator does not work, then the food is spoiled.  So if the electric power is cut, then the food is spoiled.

14. Law of Detachment  Also known as Modus Ponens  If P Q is true and P is true, then Q must be true.  For example  If an angle is obtuse, then it cannot be acute.  Angle A is obtuse.  Therefore, Angle A cannot be acute.

15. Law of Syllogism  Transitive Property

16. Is this valid?  Marcos wrote the following argument:  If the soccer team loses, Denise won’t watch their next game.  Denise watched their next game.  Therefore, the soccer team lost.  No. But what can Marcos conclude?  The soccer team won.

17. Is this valid?  Jessica wrote the following argument:  If the sun is out, then Karina will go to the beach.  If she does not go with friends, then Karina will not go to the beach.  The sun is out.  Therefore, Karina goes with friends.

18. Any time you use logical evidence, you are using deductive reasoning!

19. Review  Inductive or Deductive?  Inductive

20. Review  Inductive or Deductive?  Inductive

21. Review  Inductive or Deductive?  Inductive

22. Practice WW  True  Therefore, it will not start.

23. Deductive Reasoning  Invalid.  A, B, and C could all lie in plane G and still be collinear.

24. Deductive Reasoning  Valid.  Uses the Law of Detachment.

25. Deductive Reasoning WW  Therefore, If you get a job, then you will buy a car.

26. Practice WW BB  Law of Syllogism (Transitive Property)

27. Practice WW  Therefore, school will be closed.  Law of Detachment

28. Practice WW  Therefore MA = MB.  Law of Syllogism.

29. Challenge WW

30. Practice

31. Today’s Objectives  Explain the laws used in the deductive reasoning process.  Use deductive reasoning to lead to accurate conclusions.  Use the Law of Detachment  Use the Law of Syllogism  Use Problem Solving Skills

32. Exit Slip Retake For #1, fill in both blanks and explain your reasoning. 1.Using the Law of ____________, what can be deduced? If you check your email, you must have internet access. Michael checks his email. Therefore, _______________________________. 2.Describe the following properties: A.Reflexive B.Symmetric C.Transitive 3.Five girls took part in a race. Ana finished before Blanca but behind Concepcion. Daysi finished before Elizabeth but behind Blanca. What was the finishing order? 4.Explain your reasoning for #3. Include which law or property you used.