1. “Education is the most powerful weapon which you can use to change the world.” ― Nelson Mandela Do NowNelson Mandela  Take two sticky notes from the bookcase at the front of the room—one small one and one large one.  Take the small one and write down your name, and the day, not month, of your birthday (just write 20 if your birthday is October 20)  Stick it to the birthday chart on the correct month, stacking up to form a bar graph.  On the large one, write down your name and the name of the college(s) you would like to go to.  Stick it on the wall under the sign “College Wall”

2. Problem Solving at an early age

3. You be the judge Judging the validity of conditional statements

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

5. Rewrite in conditional (“if-then”) form  A triangle has, at most, one right angle.  If it’s a triangle, it has, at most, one right angle.  Negation?  If it’s a triangle, it has more than one right angle.

6. 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!

7. 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

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

9. 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.

10. 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”

11. 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.

12. 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.

13. Law of Syllogism  Transitive Property

14. 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.

15. 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.

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