Download Jessica wrote the following argument: If the sun is out, then Karina

“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.