“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”

Problem Solving at an early age

You be the judge Judging the validity of conditional statements

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.

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.

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!

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

Equivalence Properties  Reflexive Property  Symmetric Property  Transitive Property

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.

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”

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.

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.

Law of Syllogism  Transitive Property

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.

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.

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