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An Introduction to Logic
2.2
Why Logic?
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A proof of any form requires logical reasoning.
Logical reasoning ensures that the conclusions
you reach are true – as long as the rest of the
statements in the argument are also true.
Note: A statement is a sentence that is either true
or false.
Using Logical Reasoning

Example Statement:
– All Corvettes are Chevrolets.
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This fact can be represented by
the Euler (pronounced “oiler”)
diagram at the right.
Euler diagrams are also called
Venn diagrams.
More about Euler diagrams
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From the Euler diagram, we
can also write an “if-then”
statement.
If a car is a Corvette, then it is
a Chevrolet.
These “If-then” statements are
called conditional statements.
If ….. Then …..

In logical notation, conditionals are
written as follows:
If p then q
or
p→q
(read as “p implies q”)
By phrasing a conjecture as an if-then statement,
you can quickly identify its hypothesis and
conclusion.
Writing a Conditional Statement
Write a conditional statement from the following.
If an animal is a blue
jay, then it is a bird.
The inner oval represents the hypothesis, and
the outer oval represents the conclusion.
More about Conditionals
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In a conditional, the part following the word
if is the hypothesis and the part following
the word then is the conclusion.
Identify the hypothesis and conclusion of the
following conditional.
– If a car is a Corvette, then it is a Chevrolet.
Hypothesis
Conclusion
Example

Write the following statement as a
conditional. Underline the hypothesis
and circle the conclusion. Also draw an
Euler diagram for the statement.
“All snakes are reptiles.” Reptiles
If an animal is a snake, then it is a reptile.
hypothesis
conclusion
Answer: Hypothesis: an animal is a snake
Conclusion: it is a reptile
Snakes
Extending Euler Diagrams

Now consider the following statement:
– I have a snake named Max.
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By placing the Max into our Euler diagram,
what can you logically conclude?
● Max
Example:
Identify the hypothesis and conclusion of the
following statement.
If a polygon has 6 sides, then it is a hexagon.
If a polygon has 6 sides, then it is a hexagon.
hypothesis
conclusion
Answer: Hypothesis: a polygon has 6 sides
Conclusion: it is a hexagon
Identify the hypothesis and conclusion of each
statement.
a. If you are a baby, then you will cry.
Answer: Hypothesis: you are a baby
Conclusion: you will cry
b. To find the distance between two points, you can use
the Distance Formula.
Answer: Hypothesis: you want to find the distance
between two points
Conclusion: you can use the Distance Formula
If-Then Statements
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As we just witnessed, some conditionals are not
written in “if-then” form but in standard form. By
identifying the hypothesis and conclusion of a
statement, we can translate the statement to “ifthen” form for a better understanding.
When writing a statement in “if-then” form,
identify the requirement (condition) to find your
hypothesis and the result as your conclusion.
Example:
Identify the hypothesis and conclusion of the
following statement. Then write the statement
in the if-then form.
Distance is positive.
Sometimes you must add information to a statement.
Here you know that distance is measured or determined.
Answer: Hypothesis: a distance is determined
Conclusion: it is positive
If a distance is determined, then it is positive.
Example:
Identify the hypothesis and conclusion of the
following statement. Then write the statement
in the if-then form.
A five-sided polygon is a pentagon.
Answer: Hypothesis: a polygon has five sides
Conclusion: it is a pentagon
If a polygon has five sides, then it is
a pentagon.
Identify the hypothesis and conclusion of each
statement. Then write each statement in if-then form.
a. A polygon with 8 sides is an octagon.
Answer: Hypothesis: a polygon has 8 sides
Conclusion: it is an octagon
If a polygon has 8 sides, then it is an octagon.
b. An angle that measures 45º is an acute angle.
Answer: Hypothesis: an angle measures 45º
Conclusion: it is an acute angle
If an angle measures 45º, then it is an acute
angle.
Logical Argument
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1.
2.
3.
The complete process of drawing the conclusion
that the copperhead is a reptile can be written as
a logical argument. This particular argument has
three parts and is known as a syllogism
(detachment).
If an animal is a snake, then it is a reptile.
A copperhead is a snake.
Therefore, a copperhead is a reptile.
Logical Argument
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The process of drawing logically certain conclusions by
using an argument is known as deductive reasoning.
Another example of a logical argument:
– If an angle measures 90°, then it is a right angle.
(General statement of fact)
– Angle ABC has a measure of 90°.
(Specific statement of fact)
-- Therefore, angle ABC is a right angle.
(Logical conclusion)
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If a triangle is equilateral, then the
triangle is isosceles.
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Triangle ABC is equilateral.
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Therefore, triangle ABC is isosceles.
Changing Conditionals
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Converse
– Switch
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Inverse
– Negate
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Contrapositive
– Switch and Negate
Converse, Inverse, and Contrapositive
Statement
Formed by
Symbols
Examples
Conditional
Given an
hypothesis and
conclusion
p
q
If 2 angles have the
same measure, then
they are congruent.
Converse
Exchange the
hypothesis and
conclusion
q
p
If 2 angles are
congruent, then they
have the same measure.
Inverse
Negate both the ~p
hypothesis and
the conclusion
~q
If 2 angles do not have
the same measure, then
they are not congruent.
Contrapositive
Negate both the ~q
hypothesis and
conclusion of
the converse
~p
If 2 angles are not
congruent, then they do
not have the same
measure.
Converse
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When you switch the hypothesis and
conclusion of a conditional statement, you
have the converse of the conditional.
Example:
– Write the converse of the conditional.
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Conditional: If you have a cat, then you have a pet.
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Converse: If you have a pet, then you have a cat.
Inverse
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When you negate (not) the hypothesis and
conclusion of a conditional statement, you
have the inverse of the conditional.
Example:
– Write the inverse of the conditional
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Conditional: If you have a cat, then you have a pet.
Inverse: If you do not have a cat, then you do not
have a pet.
Contrapositive
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When you switch AND negate the
hypothesis and conclusion of a conditional
statement, you have the contrapostive of
the conditional.
Example:
– Write the contrapositive of the conditional
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Conditional: If you have a cat, then you have a pet.
Contrapositive: If you do not have a pet, then you do
not have a cat.
Truth or Lies?
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In the examples above, the conditional
statement is true. Are the related conditionals
true?
T
F
F
Conditional: If you have a cat, then you have a pet.
Converse: If you have a pet, then you have a cat.
Inverse: If you do not have a cat, then you do not have a pet.
T Contrapositive: If you do not have a pet, then you do not have a cat.
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How did you know?
Truth or Lies?
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The contrapositive of a true conditional is
always true, and the contrapositive of a
false conditional is always false.
The converse and inverse of a conditional
are either both true or both false.
An example which proves that a statement
is false is called a counterexample.

Write the converse, inverse, and
contrapositive for the following
conditional. Determine if the
statements are true or false. If false,
give a counterexample.
– If you are 16 years old, then you are a
teenager.
Conditional
If you are 16 years old, then you are a teenager.
Converse
If you are a teenager, then you are 16 years old.
Inverse
If you are not 16 years old, then you are not a teenager.
Contrapositive
If you are not a teenager, then you are not 16 years old.
Write the converse, inverse, and contrapositive of the
statement All squares are rectangles. Determine
whether each statement is true or false. If a statement
is false, give a counterexample.
First, write the conditional in if-then form.
Conditional: If a shape is a square, then it is a rectangle.
The conditional statement is true.
Converse: If a shape is a rectangle, then it is a square.
The converse is false. A rectangle with l = 2
and w = 4 is not a square.
The inverse is the negation of the hypothesis and conclusion.
Inverse:
If a shape is not a square, then it is not a
rectangle. The inverse is false. A 4-sided
polygon with side lengths 2, 2, 4, and 4 is
not a square, but it is a rectangle.
The contrapositive is the negating and switching the
hypothesis and conclusion.
Contrapositive: If a shape is not a rectangle, then it is
not a square. The contrapositive is true.
Write the converse, inverse, and contrapositive of the
statement The sum of the measures of two
complementary angles is 90. Determine whether each
statement is true or false. If a statement is false, give a
counterexample.
Answer: Conditional: If two angles are complementary, then
the sum of their measures is 90; true.
Converse: If the sum of the measures of two
angles is 90, then they are complementary; true.
Inverse: If two angles are not complementary,
then the sum of their measures is not 90; true.
Contrapositive: If the sum of the measures of
two angles is not 90, then they are not
complementary; true.
Assignment
Geometry:
2.2 Inverse and Contrapositive
Worksheet
Logical Chains
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Conditional statements that can be
linked together are called logical
chains.
Logical Chains (Example)
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Arrange the following conditionals into
a logical chain
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Given:
– 1) If it is July 4th, then flags are flying.
– 2) If flags are flying, then there is a parade.
– 3) If there is a parade, then fireworks will go off.
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Prove:
– If it is July 4, then fireworks will go off.
If-Then Transitive Property
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Logical chains rely on the following
property:
Given:
– If A then B, and if B then C.
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Conclusion:
– If A then C
Assignment
Geometry:
2.2 A Number 7
2.2B and Section 9 - 33
2.3 Definitions
BICONDITIONAL
If the original conditional is true AND the
converse is true, then the statement is a
definition.
A biconditional can be written for a definition
 Notation: p  q or p  q
 (note the double headed arrow)
 We say: “p if and only if q”
 This can be abbreviated to: p iff q

When a conditional statement AND the converse are BOTH TRUE, this
creates a special case called ‘biconditional”.
Conditional:
If a quadrilateral has 4 right angles, then it is a rectangle.
ab
(true)
Converse:
If it is a rectangle, then it is a quadrilateral with 4 right angles. ba
(true)
Biconditional:
A quadrilateral has 4 right angles if and only if it is a rectangle.
(don’t use if and then) a  b (true BOTH ways)
iff means “if and only if”
A biconditional is a DEFINITION.
A biconditional is a statement that is true backwards and forwards.
•Definitions can be written in “if
and only if” form.
•The converse of a definition is
always true.
•This means the definition is true
forwards & backwards.
Biconditional Statement
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Contains the phrase “if & only if”
Abbreviation iff
It is a conditional statement & its converse all in
one.
Ex: 3 points are collinear iff they are on the
same line.
Conditional: If 3 pts are collinear, then they
are on the same line.
Converse: If 3 pts are on the same line, then
they are collinear.
Remember:
A biconditional is written only if the
original conditional AND its
converse are true, and then it can
be called a definition.
Example:
Is the statement a biconditional?
Conditional?
If x = 3, then x2 = 9. True
Converse?
If x2 = 9, then x = 3.
Not a Biconditional
False
A conditional statement & its
converse can be rewritten as a
biconditional only if both
statements are true.
Example: Determine whether the statements
are true. If so, rewrite as one biconditional
statement.
Conditional: If a number ends in 0 or 5,
then it is evenly divisible by 5. True
Converse: If a number is evenly divisible by 5,
then it ends in 0 or 5. True
Biconditional (definition)
A number ends in 0 or 5 iff it is evenly
divisible by 5.
a
b
c
d
Floppers
e
Which ones are
Let’s write a definition: Not Floppers
Floppers?
Step 1: Write a conditional statement:
If a figure is a Flopper, then it has one eye and two tails. (true)
Step 2: Write the converse:
If a figure has one eye and two tails, then it is a Flopper. (true)
Step 3: Write the biconditional (definition)
A figure is a Flopper if and only if it has one eye and two tails.
Which of these figures are hexagons? Write a definition for hexagon.
Identify the underlined portion
of the conditional statement.
A.
B.
C.
Hypothesis
Conclusion
Neither
Identify the underlined portion of
the conditional statement.
A.
B.
C.
Hypothesis
Neither
Conclusion
Identify the underlined portion of
the conditional statement.
A.
B.
C.
Hypothesis
Conclusion
Neither
Identify the converse for the given
conditional.
A.
B.
C.
D.
If you do not like tennis, then you do
not play on the tennis team.
If you play on the tennis team, then you
like tennis.
If you do not play on the tennis team,
then you do not like tennis.
You play tennis only if you like tennis.
Identify the inverse for the given
conditional.
A.
B.
C.
D.
If
If
If
If
2x is not even, then x is not odd.
2x is even, then x is odd.
x is even, then 2x is odd.
x is not odd, then 2x is not even.
Write the conditional, converse, inverse, and
contrapositive of the conditional statement.
An adult insect is an animal that has six legs.
Conditional: If an animal is an adult insect, then it
has six legs.
Converse: If an animal has six legs, then it is an adult
insect.
Inverse: If an animal is not an adult insect, then it does
not have six legs.
Contrapositive: If an animal does not have six legs,
then it is not an adult insect.
Write the converse, inverse, and
contrapositive of the conditional statement.
“If Maria’s birthday is February 29,
then she was born in a leap year.”
Converse: If Maria was born in a leap year,
then her birthday is February 29.
Inverse: If Maria’s birthday is not February
29, then she was not born in a leap year.
Contrapositive: If Maria was not born in a
leap year, then her birthday is not
February 29.
Chevrolets
Corvettes
Write a conditional for the Euler diagram.
If a car is a Corvette, then it is a Chevrolet.
Place the logical chain in order and
find the conclusion.
 Consider this:
 If cats freak, then mice frisk.
 If sirens shriek, then dogs howl.
 If dogs howl, then cats freak.
If sirens shriek, then dogs howl.
If dogs howl, then cats freak.
If cats freak, then mice frisk.
CONCLUSION:
If sirens shriek, then mice frisk.

Rearrange the statements to create a logical chain. Then write the
proven conclusion.
–
–
–
–
1. If you go to a movie, then you will spend all of your money.
2. If you clean your room, then you will go to a movie.
3. If you cannot buy gas for the car, then you will be stranded.
4. If you spend all of your money, then you cannot buy gas for the car.
– Logical Chain Order:
2, 1, 4, 3
– Conclusion: If you clean your room, then you will be stranded.
Write a definition for polygon.
Let’s write a definition:
Step 1: Write a conditional statement:
If a figure is a polygon, then it is a closed plane figure formed by three or more
line segments. (true)
Step 2: Write the converse:
If a figure is a closed plane figure formed by three or more line segments, then
it is a polygon. (true)
Step 3: Write the biconditional (definition)
A figure is a polygon if and only if it is a closed plane figure formed
by three or more line segments.
Assignment
Geometry:
2.3B and Section 8 - 27