Explain in your own words what deductive reasoning is.

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Education is the most powerful weapon which you can use to change the world. Nelson MandelaDo NowExplain in your own words what deductive reasoning is.Exit Slip Error AnalysisCompare your exit slip to this one.Problem Solving at an early age

Deductive Reasoning IIJudging the validity of conditional statements

Todays ObjectivesExplain the laws used in the deductive reasoning process.Use deductive reasoning to lead to accurate conclusions.Use the Law of DetachmentUse the Law of SyllogismUse Problem Solving SkillsRewrite in conditional (if-then) formAll quadrilaterals have four sides.If its a quadrilateral, then it has four sides.Inverse?If its 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) formA triangle has, at most, one right angle.If its a triangle, then it has, at most, one right angle.Inverse?If its not a triangle, then it has more than one right angle.Can I just say At least? No, because a triangle can have at most and at least one right angle. We need the conditional and the negation to be mutually exclusive. Two things that cannot both happen at the same time.7Rewrite in conditional (if-then) formTwo lines in a plane always intersect at exactly one pointIf there are two lines in a plane, they intersect at exactly one pointNegation?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!Remember a counterexample fits the hypothesis but not the conclusion. So its an example of the negation. And if theres an example of the negation (even one), then the conditional statement cannot be true. The conditional and the negation cannot both be true!8Some termsAxiom a self-evident truth that requires no proof; a statement accepted as factPostulate a proposition that requires no proofTheorem 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

The proofs we do in geometry will be proving theorems, based on axioms.9Equivalence PropertiesReflexive PropertySymmetric PropertyTransitive PropertyReflexive PropertyA=AA quantity is equal to itselfIn logic, A A.Always true in logicIf youre a student at Simon Tech, then youre a student at Simon Tech.If a pentagon has five sides, then a pentagon has five sides.

Symmetric PropertyIf A=B then B=AAlways 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

For example, 12Transitive PropertyAlso known as the Law of SyllogismIf A=B and B=C then A=CIn 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 DetachmentAlso known as Modus Ponens If P Q is true and P is true, then Q must be true.For exampleIf 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 wont 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!ReviewInductive or Deductive?

Inductive

ReviewInductive or Deductive?

Deductive

ReviewInductive or Deductive?

Deductive

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TrueTherefore, it will not start.

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

Deductive ReasoningValid. Uses the Law of Detachment.

Deductive ReasoningW

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

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BLaw of Syllogism (Transitive Property)

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Therefore, school will be closed.Law of Detachment

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Therefore MA = MB.Law of Syllogism.

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PracticeTodays ObjectivesExplain the laws used in the deductive reasoning process.Use deductive reasoning to lead to accurate conclusions.Use the Law of DetachmentUse the Law of SyllogismUse Problem Solving SkillsExit SlipFor #1, fill in both blanks and explain your reasoning.Using the Law of ____________, what can be deduced? If you check your email, you must have internet access. Michael checks his email. Therefore, _______________________________.Describe the following properties: ReflexiveSymmetricTransitiveFive 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?Explain your reasoning for #3. Include which law or property you used.