“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.

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

Problem Solving at an early age

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

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

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.

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.

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!

Review  Inductive or Deductive?  Inductive

Review  Inductive or Deductive?  Inductive

Review  Inductive or Deductive?  Inductive

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

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

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

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

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

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

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

Challenge WW

Practice

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

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 , you must have internet access. Michael checks his . 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.