Logic
Eleanor Roosevelt High School
Geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
Sentences, Statements,
and Truth Values
Mr. Chin-Sung Lin
ERHS Math Geometry
Logic
Logic is the science of reasoning
The principles of logic allow us to determine if
a statement is true, false, or uncertain on
the basis of the truth of related statements
Mr. Chin-Sung Lin
ERHS Math Geometry
Sentences and Truth Values
When we can determine that a statement is
true or that it is false, that statement is said
to have a truth value
Statements with known truth values can be
combined by the laws of logic to determine
the truth value of other statements
Mr. Chin-Sung Lin
ERHS Math Geometry
Mathematical Sentences
Simple declarative statements that state a
fact, and that fact can be true or false
• Parallel lines are coplanar
TRUE
• Straight angle is 180o
TRUE
• x + (-x) = 1
FALSE
• Obtuse triangle has 2 obtuse angles
FALSE
Mr. Chin-Sung Lin
ERHS Math Geometry
Nonmathematical Sentences
Sentences that do not state a fact, such as
questions, commands, phrases, or
exclamations
• Is geometry hard?
Question
• Straight angle is 180o
Command
• All the isosceles triangles
Phrase
• Wow!
Exclamation
Mr. Chin-Sung Lin
ERHS Math Geometry
Nonmathematical Sentences
We will not discuss sentences that are true for
some persons and false for others
• I love winter
• Basket ball is the best sport
• Triangle is the most beautiful geometric shape
Mr. Chin-Sung Lin
ERHS Math Geometry
Open Sentences
Sentences that contain a variable
The truth vale of the open sentence depends on
the value of the variable
• AB = 20
Variable: AB
• 2x + 3 = 15
Variable: x
• He got 95 in geometry test
Variable: he
Mr. Chin-Sung Lin
ERHS Math Geometry
Open Sentences
Domain or Replacement Set
The set of all elements that are possible
replacements for the variable
Solution Set or Truth Set
The element(s) from the domain that make the
open sentence true
Mr. Chin-Sung Lin
ERHS Math Geometry
Solution Set or Truth Set
Example:
Open sentence:
Variable:
Domain:
Solution set:
x + 5 = 10
x
all real numbers
5
Mr. Chin-Sung Lin
ERHS Math Geometry
Solution Set or Truth Set
Example:
Open sentence:
Variable:
Domain:
Solution set:
x (1/x) = 10
x
all real numbers
Φ, { }, or empty set
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise
Identify each of the following sentences as
true, false, open, or nonmathematical
• Add  A and  B
NONMATH
• Congruent lines are always parallel
FALSE
• 3(x – 2) = 2(x – 3) + x
TRUE
• y – 6 = 2y + 7
OPEN
• Is ΔABC an equilateral triangle?
NONMATH
• Distance between 2 points is positive TRUE
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise
Use the replacement set {3, 3.14, √3, 1/3,
3π} to find the truth set of the open sentence
“It is a rational number.”
Truth Set: {3, 3.14, 1/3}
Mr. Chin-Sung Lin
ERHS Math Geometry
Statements and Symbols
A sentence that has a truth value is called a
statement or a closed sentence
Truth value can be true [T] or false [F]
In a statement, there are no variables
Mr. Chin-Sung Lin
ERHS Math Geometry
Negations
The negation of a statement always has the
opposite truth value of the original
statement and is usually formed by adding
the word not to the given statement
• Statement Right angle is 90o
TRUE
• Negation
FALSE
Right angle is not 90o
• Statement Triangle has 4 sides
• Negation Triangle does not have 4 sides
FALSE
TRUE
Mr. Chin-Sung Lin
ERHS Math Geometry
Logic Symbols
The basic element of logic is a simple
declarative sentence
We represent this element by a lowercase
letter (p, q, r, and s are the most common)
• Statement Right angle is 90o
TRUE
• Negation
FALSE
Right angle is not 90o
• Statement Triangle has 4 sides
• Negation Triangle does not have 4 sides
FALSE
TRUE
Mr. Chin-Sung Lin
ERHS Math Geometry
Logic Symbols
The basic element of logic is a simple
declarative sentence
We represent this element by a lowercase
letter (p, q, r, and s are the most common)
Mr. Chin-Sung Lin
ERHS Math Geometry
Logic Symbols
For example,
Statement p represents
Right angle is 90o
Negation ~p represents
Right angle is not 90o
~p is read “not p”
Mr. Chin-Sung Lin
ERHS Math Geometry
Logic Symbols
Symbol
P
Statement
Truth value
There are 3 sides in a triangle
T
There are not 3 sides in a triangle
F
q
2x + 3 = 2x
F
~q
2x + 3 ≠ 2x
T
r
NYC is a city
T
NYC is not a city
F
~p
~r
Mr. Chin-Sung Lin
ERHS Math Geometry
Logic Symbols
Symbol
r
~r
~(~r)
Statement
Truth value
NYC is a city
T
NYC is not a city
F
It is not true that NYC is not a city
T
T
~(~r) always has the same truth value as r
~r
~(~r)
NYC is not a city
F
NYC is a city
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table
The relationship between a statement p and its
negation ~p can be summarized in a truth table
A statement p and its negation ~p have opposite
truth values
p
~p
T
F
F
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
Mr. Chin-Sung Lin
ERHS Math Geometry
Compound Sentences / Statements
Mathematical sentences formed by connectives
such as and and or
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
A compound statement formed by combining
two simple statements using the word and
Each of the simple statements is called a
conjunct
Statement:
Conjunction
Symbols:
p, q
p and q
p^q
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
Example:
p:
A week has 7 days (T)
q:
A day has 24 hours (T)
p^q: A week has 7 days and a day has 24 hours (T)
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
A conjunction is true when both statements are
true
When one or both statements are false, the
conjunction is false
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
Example:
p:
A week has 7 days (T)
q:
A day does not have 24 hours (F)
p^q: A week has 7 days and a day does not have 24
hours (F)
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
Tree Diagram
q is true
p ^ q is true
q is false
p ^ q is false
q is true
p ^ q is false
q is false
p ^ q is false
p is true
p is false
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
Truth Table
p
q
p^q
T
T
T
T
F
F
F
T
F
F
F
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
Example:
p:
3 is an odd number (T)
q:
4 is an even number (T)
p^q:
3 is an odd number and 4 is an even
number (T)
p
q
p^q
T
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
A conjunction may contain a statement and a
negation at the same time
p
q
~q
p ^ ~q
T
T
F
F
T
F
T
T
F
T
F
F
F
F
T
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
Example:
p:
3 is an odd number (T)
q:
5 is an even number (F)
p^~q: 3 is an odd number and 5 is not an even
number (T)
p
q
~q
p ^ ~q
T
F
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
A conjunction may contain a statement and a
negation at the same time
p
q
~p
~p ^ q
T
T
F
F
T
F
F
F
F
T
T
T
F
F
T
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
Example:
p:
2 is an odd number (F)
q:
4 is an even number (T)
~p^q: 2 is not an odd number and 4 is an even
number (T)
p
q
~p
~p ^ q
F
T
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
A conjunction may contain two negations at the
same time
p
q
~p
~q
~p ^ ~q
T
T
F
F
F
T
F
F
T
F
F
T
T
F
F
F
F
T
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Conjunctions
Example:
p:
2 is an odd number (F)
q:
5 is and even number (F)
~p^~q: 2 is not an odd number and 5 is not an even
number (T)
p
q
~p
~q
~p ^ ~q
F
F
T
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
A compound statement formed by combining
two simple statements using the word or
Each of the simple statements is called a
disjunct
Statement:
Disjunction
Symbols:
p, q
p or q
pVq
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Example:
p:
A week has 7 days (T)
q:
A day has 20 hours (F)
pVq: A week has 7 days or a day has 20 hours (T)
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
A disjunction is true when one or both
statements are true
When both statements are false, the disjunction
is false
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Example:
p:
A week has 8 days (F)
q:
A day does not have 24 hours (F)
pVq: A week has 8 days or a day does not have 24
hours (F)
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Tree Diagram
q is true
p V q is true
q is false
p V q is true
q is true
p V q is true
q is false
p V q is false
p is true
p is false
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Truth Table
p
q
pVq
T
T
T
T
F
T
F
T
T
F
F
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Example:
p:
3 is an odd number (T)
q:
5 is an even number (F)
pVq:
3 is an odd number or 5 is an even
number (T)
p
q
pVq
T
F
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
A disjunction may contain a statement and a
negation at the same time
p
q
~q
p V ~q
T
T
F
T
T
F
T
T
F
T
F
F
F
F
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Example:
p:
3 is an odd number (T)
q:
5 is an even number (F)
pV~q: 3 is an odd number or 5 is not an even
number (T)
p
q
~q
p V ~q
T
F
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
A disjunction may contain a statement and a
negation at the same time
p
q
~p
~p V q
T
T
F
T
T
F
F
F
F
T
T
T
F
F
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Example:
p:
2 is an odd number (F)
q:
4 is an even number (T)
~pVq: 2 is not an odd number or 4 is an even
number (T)
p
q
~p
~p V q
F
T
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
A disjunction may contain two negations at the
same time
p
q
~p
~q
~p V ~q
T
T
F
F
F
T
F
F
T
T
F
T
T
F
T
F
F
T
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Example:
p:
2 is an odd number (F)
q:
5 is an even number (F)
~pV~q: 2 is not an odd number or 5 is not an even
number (T)
p
q
~p
~q
~p V ~q
F
F
T
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Use the following statements:
Let k represent “Kurt plays baseball.”
Let a represent “Alicia plays baseball.”
Let n represent “Nathan plays soccer.”
Write each given sentence in symbolic form:
a. Kurt or Alicia play baseball
b. Kurt plays baseball or Nathan plays soccer
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Use the following statements:
Let k represent “Kurt plays baseball.”
Let a represent “Alicia plays baseball.”
Let n represent “Nathan plays soccer.”
Write each given sentence in symbolic form:
a. Kurt or Alicia play baseball (k V a)
b. Kurt plays baseball or Nathan plays soccer (k V n)
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Use the following statements:
Let k represent “Kurt plays baseball.”
Let a represent “Alicia plays baseball.”
Let n represent “Nathan plays soccer.”
Write each given sentence in symbolic form:
a. Alicia plays baseball or Alicia does not play baseball
b. It is not true that Kurt or Alicia play baseball
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Use the following statements:
Let k represent “Kurt plays baseball.”
Let a represent “Alicia plays baseball.”
Let n represent “Nathan plays soccer.”
Write each given sentence in symbolic form:
a. Alicia plays baseball or Alicia does not play baseball
(a V ~a)
b. It is not true that Kurt or Alicia play baseball (~(k V a))
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Use the following statements:
Let k represent “Kurt plays baseball.”
Let a represent “Alicia plays baseball.”
Let n represent “Nathan plays soccer.”
Write each given sentence in symbolic form:
a. Either Kurt does not play baseball or Alicia does not
play baseball
b. It’s not the case that Alicia or Kurt play baseball
Mr. Chin-Sung Lin
ERHS Math Geometry
Disjunctions
Use the following statements:
Let k represent “Kurt plays baseball.”
Let a represent “Alicia plays baseball.”
Let n represent “Nathan plays soccer.”
Write each given sentence in symbolic form:
a. Either Kurt does not play baseball or Alicia does not
play baseball (~k V ~a)
b. It’s not the case that Alicia or Kurt play baseball
(~ (a V k))
Mr. Chin-Sung Lin
ERHS Math Geometry
Inclusive OR vs. Exclusive OR
When we use the word or to mean that one or
both of the simple sentences are true, we call this
the inclusive or
When we use the word or to mean that one and
only one of the simple sentences is true, we call
this the exclusive or
In the exclusive or, the disjunction p or q will be
true when p is true, or when q is true, but not both
Mr. Chin-Sung Lin
ERHS Math Geometry
Exclusive OR
Truth Table
p
q
p⊕q
T
T
F
T
F
T
F
T
T
F
F
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Example
Find the solution set of each of the following if the
domain is the set of positive integers less than 8
a. (x < 4) ∨ (x > 3)
b. (x > 3) ∨ (x is odd)
c. (x > 5) ∧ (x < 3)
Mr. Chin-Sung Lin
ERHS Math Geometry
Example
Find the solution set of each of the following if the
domain is the set of positive integers less than 8
a. (x < 4) ∨ (x > 3)
{1, 2, 3, 4, 5, 6, 7}
b. (x > 3) ∨ (x is odd)
{1, 3, 4, 5, 6, 7}
c. (x > 5) ∧ (x < 3)
{}
Mr. Chin-Sung Lin
ERHS Math Geometry
Conditionals
Mr. Chin-Sung Lin
ERHS Math Geometry
Conditionals (or Implications)
A compound statement formed by using the word
if…..then to combine two simple statements
Statement:
Conditional:
Symbols:
p, q
if p then q
p implies q
p only if q
pq
Mr. Chin-Sung Lin
ERHS Math Geometry
Conditionals
Example:
p:
It is raining
q:
The street is wet
pq: If it is raining then the road is wet
qp: If the street is wet then it is raining
* when we change the order of two statements in conditional,
we may not have the same truth value as the original
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts of a Conditional Statement
A conditional statement is a logical statement
that has two parts: a hypothesis (premise,
antecedent) and a conclusion (consequent)
Hypothesis
Conclusion
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts of a Conditional Statement
A conditional statement is a logical statement
that has two parts: a hypothesis (premise,
antecedent) and a conclusion (consequent)
Hypothesis
Conclusion
an assertion or a sentence
that begins an argument
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts of a Conditional Statement
A conditional statement is a logical statement
that has two parts: a hypothesis (premise,
antecedent) and a conclusion (consequent)
Hypothesis
Conclusion
the part of a sentence
that closes an argument
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts of a Conditional Statement
When a conditional statement is in if-then form,
the if part contains the hypothesis and the then
part contains the conclusion.
IF
Hypothesis
THEN Conclusion
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts of a Conditional Statement
Example:
If two angles form a linear pair, then these angles are
supplementary
IF
ΔABC is
equiangular
Hypothesis
THEN
one of the
angles is 60o
Conclusion
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts of a Conditional Statement
IF
ΔABC is
equiangular
THEN
Hypothesis
ΔABC is
equiangular
Conclusion
IMPLIES THAT
Hypothesis
ΔABC is
equiangular
Hypothesis
one of the
angles is 60o
one of the
angles is 60o
Conclusion
ONLY IF
one of the
angles is 60o
Conclusion
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Values for the Conditional p  q
Example Case 1:
p:
It is January (T)
q:
It is winter (T)
pq: If it is January then it is winter (T)
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Values for the Conditional p  q
Example Case 2:
p:
It is January (T)
q:
It is winter (F)
pq: If it is January then it is winter (F)
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Values for the Conditional p  q
Example Case 3:
p:
It is January (F)
q:
It is winter (T)
pq: If it is January then it is winter (T)
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Values for the Conditional p  q
Example Case 4:
p:
It is January (F)
q:
It is winter (F)
pq: If it is January then it is winter (T)
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Values for the Conditional p  q
A conditional is false when a true hypothesis
leads to a false condition
In all other cases, the conditional is true
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Values for the Conditional p  q
Tree Diagram
q is true
p  q is true
q is false
p  q is false
q is true
p  q is true
q is false
p  q is true
p is true
p is false
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Values for the Conditional p  q
Truth Table
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Conditionals
Example:
p:
☐ABCD is a rectangle (F)
q:
AB // CD (T)
pq:
If ☐ABCD is a rectangle then AB // CD (?)
p
q
F
T
pq
Mr. Chin-Sung Lin
ERHS Math Geometry
Conditionals
Example:
p:
☐ABCD is a rectangle (F)
q:
AB // CD (T)
pq:
If ☐ABCD is a rectangle then AB // CD (T)
p
q
pq
F
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
When I finish my homework, I will go to sleep
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
When I finish my homework, I will go to sleep
If I finish my homework, then I will go to sleep
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
The homework is easy if I pay attention in class
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
The homework is easy if I pay attention in class
If I pay attention in class, then the homework is easy
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
Linear pairs are supplementary
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
Linear pairs are supplementary
If two angles form a linear pair, then these angles are
supplementary
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
Two right angles are congruent
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
Two right angles are congruent
If two angles are right angles, then these angles are
congruent
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
Vertical angles are congruent
Mr. Chin-Sung Lin
ERHS Math Geometry
Rewrite a Statement in If-Then Form
Vertical angles are congruent
If two angles are vertical angles, then these angles are
congruent
Mr. Chin-Sung Lin
ERHS Math Geometry
Verify a Conditional Statement
A conditional statement can be true or false
To show that a conditional statement is true, you
need to prove that the conclusion is true every
time the hypothesis is true
To show that a conditional statement is false, you
need to give only one counterexample
Mr. Chin-Sung Lin
ERHS Math Geometry
Verify a Conditional Statement
Example: If two angles are vertical angles, then these
angles are congruent
During the prove process, you can not assume that
these two angles are of certain degrees, the
proof needs to cover all the possible vertical
angle pairs
Mr. Chin-Sung Lin
ERHS Math Geometry
Conditionals, Inverses,
Converses, Contrapositives &
Biconditionals
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1.
Conditional Statement
2. Converse
3.
Inverse
4.
Contrapositive
5. Biconditionsls
Mr. Chin-Sung Lin
ERHS Math Geometry
Converse
To write the converse of a conditional statement,
exchange the hypothesis and conclusion
Statement:
If m1 = 120, then 1 is obtuse
Converse:
If 1 is obtuse, then m1 = 120
Mr. Chin-Sung Lin
ERHS Math Geometry
Inverse
To write the inverse of a conditional statement,
negate both the hypothesis and conclusion
Statement:
If m1 = 120, then 1 is obtuse
Inverse:
If m1 ≠ 120, then 1 is not obtuse
Mr. Chin-Sung Lin
ERHS Math Geometry
Contrapositive
To write the contrapositive of a conditional
statement, first write the converse, and then
negate both the hypothesis and conclusion
Statement:
If m1 = 120, then 1 is obtuse
Contrapositive:
If 1 is not obtuse, then m1 ≠ 120
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement
If m1 = 120, then 1 is obtuse
2. Converse
If 1 is obtuse, then m1 = 120
3. Inverse
If m1 ≠ 120, then 1 is not obtuse
4. Contrapositive
If 1 is not obtuse, then m1 ≠ 120
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement
If you are a basketball player, then you are an athlete
2. Converse
3. Inverse
4. Contrapositive
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement
If you are a basketball player, then you are an athlete
2. Converse
If you are an athlete, then you are a basketball player
3. Inverse
4. Contrapositive
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement
If you are a basketball player, then you are an athlete
2. Converse
If you are an athlete, then you are a basketball player
3. Inverse
If you are not a basketball player, then you are not an
athlete
4. Contrapositive
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement
If you are a basketball player, then you are an athlete
2. Converse
If you are an athlete, then you are a basketball player
3. Inverse
If you are not a basketball player, then you are not an
athlete
4. Contrapositive
If you are not an athlete, then you are not a
basketball player
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement (TRUE)
If you are a basketball player, then you are an athlete
2. Converse
If you are an athlete, then you are a basketball player
3. Inverse
If you are not a basketball player, then you are not an
athlete
4. Contrapositive
If you are not an athlete, then you are not a
basketball player
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement (TRUE)
If you are a basketball player, then you are an athlete
2. Converse (FALSE)
If you are an athlete, then you are a basketball player
3. Inverse
If you are not a basketball player, then you are not an
athlete
4. Contrapositive
If you are not an athlete, then you are not a
basketball player
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement (TRUE)
If you are a basketball player, then you are an athlete
2. Converse (FALSE)
If you are an athlete, then you are a basketball player
3. Inverse (FALSE)
If you are not a basketball player, then you are not an
athlete
4. Contrapositive
If you are not an athlete, then you are not a
basketball player
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement (TRUE)
If you are a basketball player, then you are an athlete
2. Converse (FALSE)
If you are an athlete, then you are a basketball player
3. Inverse (FALSE)
If you are not a basketball player, then you are not an
athlete
4. Contrapositive (TRUE)
If you are not an athlete, then you are not a
basketball player
Mr. Chin-Sung Lin
ERHS Math Geometry
Related Conditional Statements
1. Conditional Statement (TRUE)
If you are a basketball player, then you are an athlete
2. Converse (FALSE)
If you are an athlete, then you are a basketball player
3. Inverse (FALSE)
If you are not a basketball player, then you are not an
athlete
4. Contrapositive (TRUE)
If you are not an athlete, then you are not a
basketball player
Mr. Chin-Sung Lin
ERHS Math Geometry
Biconditional Statements
When a conditional statement and its converse are
both true, you can write them as a single
biconditional statement
A biconditional is the conjunction of a conditional
and its converse
A biconditional statement is a statement that
contains the phrase “if and only if”
Mr. Chin-Sung Lin
ERHS Math Geometry
Biconditional Statements
Statement
If two lines intersect to form a right angle, then
they are perpendicular
Converse
If two lines are perpendicular, then they
intersect to form a right angle
Bidirectional statement
Two lines are perpendicular if and only if they
intersect to form a right angle
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation
Conditional statements can be written using
symbolic notation:
Letters (e.g. p)
“statements”
Arrow ()
“implies” connects the hypothesis
and conclusion
Negation (~)
“not” negates a statement as ~p
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Conditional
Conditional Statement
If two lines intersect to form a right angle, then
they are perpendicular
Let p be “two lines intersect to form a right angle”
Let q be “they are perpendicular”
If p, then q
pq
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Converse
Conditional Statement
If two lines intersect to form a right angle, then
they are perpendicular
If p, then q
pq
Converse
If two lines are perpendicular, then they intersect
to form a right angle
If q, then p
qp
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Inverse
Conditional Statement
If two lines intersect to form a right angle, then
they are perpendicular
If p, then q
pq
Inverse
If two lines intersect not to form a right angle,
then they are not perpendicular
If not p, then not q
~p  ~q
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Contrapositive
Conditional Statement
If two lines intersect to form a right angle, then
they are perpendicular
If p, then q
pq
Contrapositive
If two lines are not perpendicular, then they
intersect not to form a right angle
If not q, then not p
~q  ~p
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Biconditional
Conditional Statement
If two lines intersect to form a right angle, then
they are perpendicular
If p, then q
pq
Biconditional
Two lines intersect to form a right angle if and
only if they are perpendicular
p if and only if q
pq
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Summary
Conditional Statement
If p, then q
pq
Converse
If q, then p
qp
If not p, then not q
~p  ~q
If not q, then not p
~q  ~p
Inverse
Contrapositive
Biconditional
p if and only if q
pq
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Exercise
Let p be “m1 = 120”, and let q be “1 is obtuse”
1. Write the p  q in words (conditional)
2. Write the q  p in words (converse)
3. Write the ~p  ~q in words (inverse)
4. Write the ~q  ~p in words (contrapositive)
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Exercise
Let p be “m1 = 120”, and let q be “1 is obtuse”
1. Write the p  q in words (conditional)
If m1 = 120, then 1 is obtuse
2. Write the q  p in words (converse)
3. Write the ~p  ~q in words (inverse)
4. Write the ~q  ~p in words (contrapositive)
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Exercise
Let p be “m1 = 120”, and let q be “1 is obtuse”
1. Write the p  q in words (conditional)
If m1 = 120, then 1 is obtuse
2. Write the q  p in words (converse)
If 1 is obtuse, then m1 = 120
3. Write the ~p  ~q in words (inverse)
4. Write the ~q  ~p in words (contrapositive)
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Exercise
Let p be “m1 = 120”, and let q be “1 is obtuse”
1. Write the p  q in words (conditional)
If m1 = 120, then 1 is obtuse
2. Write the q  p in words (converse)
If 1 is obtuse, then m1 = 120
3. Write the ~p  ~q in words (inverse)
If m1 ≠ 120, then 1 is not obtuse
4. Write the ~q  ~p in words (contrapositive)
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Exercise
Let p be “m1 = 120”, and let q be “1 is obtuse”
1. Write the p  q in words (conditional)
If m1 = 120, then 1 is obtuse
2. Write the q  p in words (converse)
If 1 is obtuse, then m1 = 120
3. Write the ~p  ~q in words (inverse)
If m1 ≠ 120, then 1 is not obtuse
4. Write the ~q  ~p in words (contrapositive)
If 1 is not obtuse, then m1 ≠ 120
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Exercise
Let p be “m1 = 90”, and let q be “1 is a right
angle”
1. Write the p  q in words (biconditional)
Mr. Chin-Sung Lin
ERHS Math Geometry
Symbolic Notation - Exercise
Let p be “m1 = 90”, and let q be “1 is a right
angle”
1. Write the p  q in words (biconditional)
m1 = 90 if and only if 1 is a right angle
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Implication
Implication: p  q
The statement “p implies q” means that if p is true,
then q must be also true
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Implication
Conditional
For hypothesis p and
conclusion q:
The condition p  q is only
false when a true
hypothesis produce a false
conclusion
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Conditional
Conditional
P: you get >90 in all tests
q: you pass the class
pq:
If you get >90 in all tests
then you pass the class
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Converse
Converse
P: you get >90 in all tests
q: you pass the class
qp:
If you pass the class
then you get >90 in all
tests
p
q
qp
T
T
T
T
F
T
F
T
F
F
F
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Inverse
Inverse
P: you get >90 in all tests
q: you pass the class
~p~q:
If you don’t get >90 in all
tests
then you don’t pass the
class
p
q
~p  ~q
T
T
T
T
F
T
F
T
F
F
F
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Contrapositive
Contrapositive
P: you get >90 in all tests
q: you pass the class
~q~p:
If you don’t pass the class
then you don’t get >90 in
all tests
p
q
~q  ~p
T
T
T
T
F
F
F
T
T
F
F
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Summary
p
q
pq
qp
~p  ~q
~q  ~p
T
T
T
T
T
T
T
F
F
T
T
F
F
T
T
F
F
T
F
F
T
T
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Summary
p
q
pq
qp
~p  ~q
~q  ~p
T
T
T
T
T
T
T
F
F
T
T
F
F
T
T
F
F
T
F
F
T
T
T
T
Equivalent
Statements
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Equivalent Statements
The conditional and the contrapositive are
equivalent statements (logical equivalents)
p q
If you get >90 in all tests, then you pass the class
~q~p
If you don’t pass the class, then you don’t get >90
in all tests
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Equivalent Statements
The converse and the inverse are equivalent
statements (logical equivalents)
q p
If you pass the class, then you get >90 in all tests
~p~q
If you don’t get >90 in all tests, then you don’t
pass the class
Mr. Chin-Sung Lin
ERHS Math Geometry
Equivalent Statements : Exercise
Write the logical equivalent for the statement “If a
polygon is a triangle, then it has three sides.”
Mr. Chin-Sung Lin
ERHS Math Geometry
Equivalent Statements : Exercise
Write the logical equivalent for the statement “If a
polygon is a triangle, then it has three sides.”
If a polygon does not have three sides, then it is not
a triangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Equivalent Statements : Exercise
Write the logical equivalent for the statement “If two
nonintersecting lines are not coplanar, then they
are skew line.”
Mr. Chin-Sung Lin
ERHS Math Geometry
Equivalent Statements : Exercise
Write the logical equivalent for the statement “If two
nonintersecting lines are not coplanar, then they
are skew line.”
If two nonintersecting lines are not skew lines, then
they are coplanar
Mr. Chin-Sung Lin
ERHS Math Geometry
Biconditionals
A biconditional is true when two statements are
both true or both false
When two statements have different truth values,
the biconditional is false
Mr. Chin-Sung Lin
ERHS Math Geometry
Truth Table - Biconditional
p
q
pq
qp
(p  q) ^ (q  p)
pq
T
T
T
T
T
T
T
F
F
T
F
F
F
T
T
F
F
F
F
F
T
T
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Applications of Biconditionals
Definitions are true biconditionals
• Right angles are angles with measure of 90
• Angles with measure of 90 are right angles
• Congruent segments are segments with the same
measure
• Segments with the same measure are congruent
segments
Mr. Chin-Sung Lin
ERHS Math Geometry
Applications of Biconditionals
Biconditionals are used to solve equations
• If x + 3 = 5, then x = 2
• If x = 2, then x + 3 = 5
* The solution of an equation is a series of biconditionals
Mr. Chin-Sung Lin
ERHS Math Geometry
Applications of Biconditionals
Biconditionals state logical equivalents
• ~(p ^ q)
(~p V ~q)
p
q
~p
~q
p^q
~(p ^ q)
~p V ~q
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Laws of Logic
Mr. Chin-Sung Lin
ERHS Math Geometry
Laws of Logic
The thought patterns used to combine the known
facts in order to establish the truth of related
facts and draw conclusions
Mr. Chin-Sung Lin
ERHS Math Geometry
Laws of Logic - Law of Detachment
Law of Detachment - Direct Argument
A valid argument uses a series of statements
called premises that have known truth values
to arrive at a conclusion
If the hypothesis of a true conditional statement
is true, then the conclusion is also true
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
If a conditional (pq) is true and the hypothesis
(p) is true, then the conclusion (q) is true
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
If two segment have the same length, then they
are congruent
You know that AB = CD
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
If two segment have the same length, then they
are congruent
You know that AB = CD
Since AB = CD satisfies the hypothesis of a true
conditional statement, the conclusion is also
true. So,
AB  CD
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
Johnson watches TV every Thursday and Saturday
night
Today is Thursday
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
Johnson watches TV every Thursday and Saturday
night
Today is Thursday
So, Johnson will watch TV tonight
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
All men will die
Mr. Lin is a man
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
All men will die
Mr. Lin is a man
So, Mr. Lin will die
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
All human will die
Mr. Lin does not die
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
All human will die
Mr. Lin does not die
So, Mr. Lin is not human
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
Vertical angles are congruent
A and C are vertical angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Detachment
Vertical angles are congruent
A and C are vertical angles
then, A  C
Mr. Chin-Sung Lin
ERHS Math Geometry
Laws of Logic - Law of Disjunctive
Inference
Law of Disjunctive Inference
When a disjunction is true and one of the
disjuncts is false, then the other disjunct
must be true
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Disjunctive Inference
If a disjunction (pVq) is true and the disjunct (p) is
false, then the other disjunct (q) is true
If a disjunction (pVq) is true and the disjunct (q) is
false, then the other disjunct (p) is true
p
q
pVq
T
T
T
T
F
T
F
T
T
F
F
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Disjunctive Inference
I walk to school or I take bus to school
I do not walk to school
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Disjunctive Inference
I walk to school or I take bus to school
I do not walk to school
So, I take bus to school
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Disjunctive Inference
Johnson watches TV every Thursday or Saturday
Johnson does not watche TV this Thursday
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Disjunctive Inference
Johnson watches TV every Thursday or Saturday
Johnson does not watch TV this Thursday
So, Johnson will watch TV this Saturday
Mr. Chin-Sung Lin
ERHS Math Geometry
Laws of Logic - Law of Syllogism
Law of Syllogism - Chain Rule
If hypothesis p, then conclusion q
If hypothesis q, then conclusion r
If these
statements
are true
If hypothesis p, then conclusion r
then this
statement
is true
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Syllogism
If two angles are linear pair, then they are
supplementary
If two angles are supplementary, then the sum of
the measure of these angles are equal to 180
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Syllogism
If two angles are linear pair, then they are
supplementary
If two angles are supplementary, then the sum of
the measure of these angles are equal to 180
If two angles are linear pair, then the sum of the
measure of these angles are equal to 180
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Syllogism
If x2 > 25, then x2 > 20
If x > 5, then x2 > 25
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Syllogism
x2
x2
If
> 25, then
> 20
If x > 5, then x2 > 25
The order of the statement
doesn’t affect the
application of the law of
syllogism
If x > 5, then x2 > 20
Mr. Chin-Sung Lin
ERHS Math Geometry
Law of Syllogism
If two triangles are congruent, then their
corresponding sides are congruent
If two triangles are congruent, then their
corresponding angles are congruent
Neither statement’s conclusion is the same as other
statement’s hypothesis. So, you cannot use law
of syllogism to write another conditional
statement
Mr. Chin-Sung Lin
ERHS Math Geometry
Drawing Conclusions
Mr. Chin-Sung Lin
ERHS Math Geometry
Drawing conclusions
The three statements given below are each true. What
conclusion can be found to be true?
1. If Rachel joins the choir then Rachel likes to sing
2. Rachel will join the choir or Rachel will play
basketball
3. Rachel does not like to sing
Mr. Chin-Sung Lin
ERHS Math Geometry
Drawing conclusions
The three statements given below are each true. What
conclusion can be found to be true?
1. If Rachel joins the choir then Rachel likes to sing
2. Rachel will join the choir or Rachel will play
basketball
3. Rachel does not like to sing
Let c represent “Rachel joins the choir”
s represent “Rachel likes to sing”
b represent “Rachel will play basketball”
Mr. Chin-Sung Lin
ERHS Math Geometry
Drawing conclusions
Original statements
1. If Rachel joins the choir then Rachel likes to sing
2. Rachel will join the choir or Rachel will play
basketball
3. Rachel does not like to sing
Convert to symbolic form
1. c  s
2. c V b
3. ~s
Mr. Chin-Sung Lin
ERHS Math Geometry
Drawing conclusions
Symbolic form
1. c  s
2. c V b
3. ~s
Draw conclusions
1. c  s is true, so ~s  ~c is true (contrapositive)
2. ~s is true, so ~c is true (law of detachment)
3. ~c is true, so c is false (negation)
4. c V b is true and c is false, so, b is true (law of
disjunctive inference)
Mr. Chin-Sung Lin
ERHS Math Geometry
Drawing conclusions
The three statements given below are each true. What
conclusion can be found to be true?
1. If Rachel joins the choir then Rachel likes to sing
2. Rachel will join the choir or Rachel will play
basketball
3. Rachel does not like to sing
Conclusion
b is true, so, “Rachel will play basketball“
Mr. Chin-Sung Lin
ERHS Math Geometry
Q&A
Mr. Chin-Sung Lin
ERHS Math Geometry
The End
Mr. Chin-Sung Lin