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Education is the most powerful weapon which you can use to
change the world. Nelson MandelaDo NowPut 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
AnalysisCompare your exit slip to this one.Problem Solving at an
early age
You be the judgeDeductive 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?Marcos wrote the following argument:If the soccer
team loses, Denise wont 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!ReviewInductive or Deductive?
Inductive
ReviewInductive or Deductive?
Inductive
ReviewInductive or Deductive?
Inductive
PracticeW
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.
PracticeW
BLaw of Syllogism (Transitive Property)
PracticeW
Therefore, school will be closed.Law of Detachment
PracticeW
Therefore MA = MB.Law of Syllogism.
ChallengeW
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 Slip RetakeFor #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.