Mr' Okenfuss

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Mr. Okenfuss
Logic Proofs
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Our Integrated II curriculum General Objective 7
states that Students will analyze the
principles of logic and compare types of proofs.
Turn to pg. 365 and lets take a look at an
example.
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Implications
All teachers have gone to college.
If a a person is a teacher, then they have gone
to college.
Every teacher has gone to college.
The fact that a person is a teacher implies that
they have gone to college.
A person is a teacher only if they go to college
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Use all four types of of conditional statements
to rewrite this statement All rectangles have
congruent diagonals.
Which of these is true?
The converse of an if p, then q statement is
if q, then p. Be careful, because the converse
of a true conditional statement may not always be
true. Conditional If it is raining, then it is
cloudy. Converse If it is cloudy, then it is
raining.
Since we can use Venn Diagrams to model
conjunctions, disjunctions and negations, do you
think that we can use them to model implications?

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Book Assignment
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Clue 1 If Don is going, then Eve is going.
Clue 2 Be is not going to the party.
Clue 3 If Al is going, then Ben is going.
Clue 4 If Carla is going, then Don is going.
Clue 5 Al or Carla is going to the party.
Questions 1. Is Ben going to the party? 2. Is
Al going to the party? 3. Is Carla going to the
party? 4. Is Don going to the party? 5. Is Eve
going to the party? 6. Which students are going?
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Types of arguments
All arguments are made of two statements, a
premise, and a conclusion. The premise is a
statement that produces the conclusion. In each
of these, the premise is the p, while the
conclusion is the q.

• Direct argument If p, then q.
• if p is true, then q must be true as well.
• Indirect argument If p, then q.
• if q is not true, then p is not true

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• Chain Rule If p, then q. If q, then r.
• if p is true, then q is true. Since q is true,
  r is true as well.
• Therefore if p is true, then r is true.
• Or rule p or q
• if p is not true, then q must be true

Anytime that both the premise and conclusion are
true, the statement is said to be a valid
argument.
An invalid argument is made when you draw a
conclusion that does not use the rules of logic.
In this case, the premise is true, but the
conclusion is false.
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Book assignment
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Biconditionals
Any true conditional that has a converse that is
also true is a biconditional statement.
Using the language of logic, we can say that p
implies q and q implies p.
Ex. If an angle is 900, then it is a right
angle If an angle is a right angle, then it is
900.
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Any definition is based on a conditional that has
a true converse. In other words, to make a good
definition, your statement should be a
biconditional statement.
Ex. A square is a rectangle with 4 congruent
sides. using if and only if A figure is a square
if and only if it is a rectangle with 4 congruent
sides.
Students, on your own paper write an if and only
if statement for this definition A
parallelogram has 2 sets of parallel sides.
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What is wrong with these attempts at definitions?
An angle is acute if the measure of the angle is
850.
The number b is a square root of a if and only if
bsquare root of a
Using symbolic logic, we can show a statement is
a biconditional similar to what we use for
implications. After all, a biconditional is an
implication that is true in both directions. p q
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Book assignment
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Proofs
A proof is where a mathematician can show that an
implication is true using one or more of the
rules of logic. The conclusion drawn from a
proof is undisputable and will forever be held as
fact!
Look at an example of a proof on pg. 394.
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Example of 2-column proof (use smart notebook)
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Example of paragraph proof
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Example of flow proof (use smart notebook)
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Algebra Proofs