Education is the most powerful weapon which you

“Education is the most powerful weapon which you can use to change the world. ” ― Nelson Mandela Do Now 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

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 A B 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? Christian wrote the following argument: If the soccer team loses, Janalee won’t watch their next game. Janalee watched their next game. Therefore, the soccer team lost. No. But what can Christian 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? Deductive

Review Inductive or Deductive? Deductive

Practice 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 Therefore, If you get a job, then you will buy a car.

Practice W B Law of Syllogism (Transitive Property)

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

Practice W Therefore MA = MB. Law of Syllogism.

Challenge 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 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.