Logic Gates & Boolean Algebra Chin-Sung Lin Eleanor Roosevelt High School.
Logic Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin.
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Transcript of Logic Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin.
Logic
Eleanor Roosevelt High School
Geometry
Mr. Chin-Sung Lin
Sentences, Statements, and Truth Values
ERHS Math Geometry
Mr. Chin-Sung Lin
Logic
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Sentences and Truth Values
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Mathematical Sentences
ERHS Math Geometry
Mr. Chin-Sung Lin
Simple declarative statements that state a fact, and that fact can be true or false
• Parallel lines are coplanar
• Straight angle is 180o
• x + (-x) = 1
• Obtuse triangle has 2 obtuse angles
TRUE
TRUE
FALSE
FALSE
Nonmathematical Sentences
ERHS Math Geometry
Mr. Chin-Sung Lin
Sentences that do not state a fact, such as questions, commands, phrases, or exclamations
• Is geometry hard?
• Straight angle is 180o
• All the isosceles triangles
• Wow!
Question
Command
Phrase
Exclamation
Nonmathematical Sentences
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Open Sentences
ERHS Math Geometry
Mr. Chin-Sung Lin
Sentences that contain a variableThe truth vale of the open sentence depends on the value of the variable
• AB = 20
• 2x + 3 = 15
• He got 95 in geometry test
Variable: AB
Variable: x
Variable: he
Open Sentences
ERHS Math Geometry
Mr. Chin-Sung Lin
The set of all elements that are possible replacements for the variable
Domain or Replacement Set
The element(s) from the domain that make the open sentence true
Solution Set or Truth Set
Solution Set or Truth Set
ERHS Math Geometry
Mr. Chin-Sung Lin
Example:
Open sentence: x + 5 = 10Variable: xDomain: all real numbersSolution set: 5
Solution Set or Truth Set
ERHS Math Geometry
Mr. Chin-Sung Lin
Example:
Open sentence: x (1/x) = 10Variable: xDomain: all real numbersSolution set: Φ, { }, or empty set
Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Identify each of the following sentences as true, false, open, or nonmathematical
• Add A and B
• Congruent lines are always parallel
• 3(x – 2) = 2(x – 3) + x
• y – 6 = 2y + 7
NONMATH
FALSE
TRUE
OPEN
• Is ΔABC an equilateral triangle?
• Distance between 2 points is positive
NONMATH
TRUE
Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
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}
Statements and Symbols
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Negations
ERHS Math Geometry
Mr. Chin-Sung Lin
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
• Negation Right angle is not 90o
TRUE
FALSE
• Statement Triangle has 4 sides• Negation Triangle does not have 4 sides
FALSE
TRUE
Logic Symbols
ERHS Math Geometry
Mr. Chin-Sung Lin
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
• Negation Right angle is not 90o
TRUE
FALSE
• Statement Triangle has 4 sides• Negation Triangle does not have 4 sides
FALSE
TRUE
Logic Symbols
ERHS Math Geometry
Mr. Chin-Sung Lin
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)
Logic Symbols
ERHS Math Geometry
Mr. Chin-Sung Lin
For example,
Statement p represents Right angle is 90o
Negation ~p represents Right angle is not 90o
~p is read “not p”
Logic Symbols
ERHS Math Geometry
Mr. Chin-Sung Lin
Symbol Statement Truth value
P There are 3 sides in a triangle T ~p 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
~r NYC is not a city F
Logic Symbols
ERHS Math Geometry
Mr. Chin-Sung Lin
Symbol Statement Truth value
r NYC is a city T ~r NYC is not a city F
~(~r) It is not true that NYC is not a city TT
~(~r) always has the same truth value as r
~r NYC is not a city F
~(~r) NYC is a city T
Truth Table
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
Compound Sentences / Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
Mathematical sentences formed by connectives such as and and or
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
A compound statement formed by combining two simple statements using the word and
Each of the simple statements is called a conjunct
Statement: p, q Conjunction p and q Symbols: p ^ q
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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)
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
A conjunction is true when both statements are true
When one or both statements are false, the conjunction is false
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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)
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
p is true
p is false
q is true
q is false
q is true
q is false
p ^ q is true
p ^ q is false
p ^ q is false
p ^ q is false
Tree Diagram
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
Truth Table
p q p ^ q
T T T
T F F
F T F
F F F
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Conjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
A compound statement formed by combining two simple statements using the word or
Each of the simple statements is called a disjunct
Statement: p, q Disjunction p or q Symbols: p V q
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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)
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
A disjunction is true when one or both statements are true
When both statements are false, the disjunction is false
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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)
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
p is true
p is false
q is true
q is false
q is true
q is false
p V q is true
p V q is true
p V q is true
p V q is false
Tree Diagram
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
Truth Table
p q p V q
T T T
T F T
F T T
F F F
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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 p V q
T F T
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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)
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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))
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Disjunctions
ERHS Math Geometry
Mr. Chin-Sung Lin
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))
Inclusive OR vs. Exclusive OR
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Exclusive OR
ERHS Math Geometry
Mr. Chin-Sung Lin
Truth Table
p q p ⊕ q
T T F
T F T
F T T
F F F
Example
ERHS Math Geometry
Mr. Chin-Sung Lin
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)∧
Example
ERHS Math Geometry
Mr. Chin-Sung Lin
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)∧ { }
Conditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
Conditionals (or Implications)
ERHS Math Geometry
Mr. Chin-Sung Lin
A compound statement formed by using the word if…..then to combine two simple statements
Statement: p, q Conditional: if p then q
p implies q p only if q
Symbols: p q
Conditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
Example:
p: It is raining
q: The street is wet
pq: If it is raining then the road is wet
qp: 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
Parts of a Conditional Statement
ERHS Math Geometry
Mr. Chin-Sung Lin
A conditional statement is a logical statement that has two parts: a hypothesis (premise, antecedent) and a conclusion (consequent)
Hypothesis Conclusion
Parts of a Conditional Statement
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Parts of a Conditional Statement
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Parts of a Conditional Statement
ERHS Math Geometry
Mr. Chin-Sung Lin
When a conditional statement is in if-then form, the if part contains the hypothesis and the then part contains the conclusion.
Hypothesis ConclusionIF THEN
Parts of a Conditional Statement
ERHS Math Geometry
Mr. Chin-Sung Lin
Example:If two angles form a linear pair, then these angles are
supplementary
ΔABC is equiangularIF THEN
one of the angles is 60o
Hypothesis Conclusion
Parts of a Conditional Statement
ERHS Math Geometry
Mr. Chin-Sung Lin
ΔABC is equiangularIF THEN
Hypothesis Conclusion
one of the angles is 60o
ΔABC is equiangular IMPLIES THAT
Hypothesis Conclusion
one of the angles is 60o
ΔABC is equiangular ONLY IF
Hypothesis Conclusion
one of the angles is 60o
Truth Values for the Conditional p q
ERHS Math Geometry
Mr. Chin-Sung Lin
Example Case 1:
p: It is January (T)
q: It is winter (T)
pq: If it is January then it is winter (T)
Truth Values for the Conditional p q
ERHS Math Geometry
Mr. Chin-Sung Lin
Example Case 2:
p: It is January (T)
q: It is winter (F)
pq: If it is January then it is winter (F)
Truth Values for the Conditional p q
ERHS Math Geometry
Mr. Chin-Sung Lin
Example Case 3:
p: It is January (F)
q: It is winter (T)
pq: If it is January then it is winter (T)
Truth Values for the Conditional p q
ERHS Math Geometry
Mr. Chin-Sung Lin
Example Case 4:
p: It is January (F)
q: It is winter (F)
pq: If it is January then it is winter (T)
Truth Values for the Conditional p q
ERHS Math Geometry
Mr. Chin-Sung Lin
A conditional is false when a true hypothesis leads to a false condition
In all other cases, the conditional is true
Truth Values for the Conditional p q
ERHS Math Geometry
Mr. Chin-Sung Lin
p is true
p is false
q is true
q is false
q is true
q is false
p q is true
p q is false
p q is true
p q is true
Tree Diagram
Truth Values for the Conditional p q
ERHS Math Geometry
Mr. Chin-Sung Lin
Truth Table
p q p q
T T T
T F F
F T T
F F T
Conditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
Example:
p: ☐ABCD is a rectangle (F)
q: AB // CD (T)
pq: If ☐ABCD is a rectangle then AB // CD (?)
p q p q
F T
Conditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
Example:
p: ☐ABCD is a rectangle (F)
q: AB // CD (T)
pq: If ☐ABCD is a rectangle then AB // CD (T)
p q p q
F T T
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
When I finish my homework, I will go to sleep
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
When I finish my homework, I will go to sleep
If I finish my homework, then I will go to sleep
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
The homework is easy if I pay attention in class
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
The homework is easy if I pay attention in class
If I pay attention in class, then the homework is easy
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
Linear pairs are supplementary
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
Linear pairs are supplementary
If two angles form a linear pair, then these angles are supplementary
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
Two right angles are congruent
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
Two right angles are congruent
If two angles are right angles, then these angles are congruent
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
Vertical angles are congruent
Rewrite a Statement in If-Then Form
ERHS Math Geometry
Mr. Chin-Sung Lin
Vertical angles are congruent
If two angles are vertical angles, then these angles are congruent
Verify a Conditional Statement
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Verify a Conditional Statement
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Conditionals, Inverses, Converses, Contrapositives &
Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
1. Conditional Statement
2. Converse
3. Inverse
4. Contrapositive
5. Biconditionsls
Converse
ERHS Math Geometry
Mr. Chin-Sung Lin
To write the converse of a conditional statement, exchange the hypothesis and conclusion
Statement:If m1 = 120, then 1 is obtuse
Converse:If 1 is obtuse, then m1 = 120
Inverse
ERHS Math Geometry
Mr. Chin-Sung Lin
To write the inverse of a conditional statement, negate both the hypothesis and conclusion
Statement:If m1 = 120, then 1 is obtuse
Inverse: If m1 ≠ 120, then 1 is not obtuse
Contrapositive
ERHS Math Geometry
Mr. Chin-Sung Lin
To write the contrapositive of a conditional statement, first write the converse, and then negate both the hypothesis and conclusion
Statement:If m1 = 120, then 1 is obtuse
Contrapositive:If 1 is not obtuse, then m1 ≠ 120
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
1. Conditional StatementIf m1 = 120, then 1 is obtuse
2. ConverseIf 1 is obtuse, then m1 = 120
3. InverseIf m1 ≠ 120, then 1 is not obtuse
4. ContrapositiveIf 1 is not obtuse, then m1 ≠ 120
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
1. Conditional StatementIf you are a basketball player, then you are an athlete
2. Converse
3. Inverse
4. Contrapositive
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
1. Conditional StatementIf you are a basketball player, then you are an athlete
2. ConverseIf you are an athlete, then you are a basketball player
3. Inverse
4. Contrapositive
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
1. Conditional StatementIf you are a basketball player, then you are an athlete
2. ConverseIf you are an athlete, then you are a basketball player
3. InverseIf you are not a basketball player, then you are not an athlete
4. Contrapositive
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
1. Conditional StatementIf you are a basketball player, then you are an athlete
2. ConverseIf you are an athlete, then you are a basketball player
3. InverseIf you are not a basketball player, then you are not an athlete
4. ContrapositiveIf you are not an athlete, then you are not a basketball player
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
1. Conditional Statement (TRUE)If you are a basketball player, then you are an athlete
2. ConverseIf you are an athlete, then you are a basketball player
3. InverseIf you are not a basketball player, then you are not an athlete
4. ContrapositiveIf you are not an athlete, then you are not a basketball player
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
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. InverseIf you are not a basketball player, then you are not an athlete
4. ContrapositiveIf you are not an athlete, then you are not a basketball player
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
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. ContrapositiveIf you are not an athlete, then you are not a basketball player
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Related Conditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Biconditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
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”
Biconditional Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
StatementIf two lines intersect to form a right angle, then they are perpendicular
ConverseIf two lines are perpendicular, then they intersect to form a right angle
Bidirectional statementTwo lines are perpendicular if and only if they intersect to form a right angle
Symbolic Notation
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Symbolic Notation - Conditional
ERHS Math Geometry
Mr. Chin-Sung Lin
Conditional StatementIf 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
Symbolic Notation - Converse
ERHS Math Geometry
Mr. Chin-Sung Lin
Conditional Statement
If two lines intersect to form a right angle, then they are perpendicular
If p, then q p q
ConverseIf two lines are perpendicular, then they intersect to form a right angle
If q, then p q p
Symbolic Notation - Inverse
ERHS Math Geometry
Mr. Chin-Sung Lin
Conditional Statement
If two lines intersect to form a right angle, then they are perpendicular
If p, then q p q
InverseIf two lines intersect not to form a right angle, then they are not perpendicular
If not p, then not q ~p ~q
Symbolic Notation - Contrapositive
ERHS Math Geometry
Mr. Chin-Sung Lin
Conditional Statement
If two lines intersect to form a right angle, then they are perpendicular
If p, then q p q
ContrapositiveIf two lines are not perpendicular, then they intersect not to form a right angle
If not q, then not p ~q ~p
Symbolic Notation - Biconditional
ERHS Math Geometry
Mr. Chin-Sung Lin
Conditional Statement
If two lines intersect to form a right angle, then they are perpendicular
If p, then q p q
BiconditionalTwo lines intersect to form a right angle if and only if they are perpendicular
p if and only if q p q
Symbolic Notation - Summary
ERHS Math Geometry
Mr. Chin-Sung Lin
Conditional StatementIf p, then q p q
ConverseIf q, then p q p
InverseIf not p, then not q ~p ~q
ContrapositiveIf not q, then not p ~q ~p
Biconditionalp if and only if q p q
Symbolic Notation - Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Let p be “m1 = 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)
Symbolic Notation - Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Let p be “m1 = 120”, and let q be “1 is obtuse”
1. Write the p q in words (conditional) If m1 = 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)
Symbolic Notation - Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Let p be “m1 = 120”, and let q be “1 is obtuse”
1. Write the p q in words (conditional) If m1 = 120, then 1 is obtuse
2. Write the q p in words (converse) If 1 is obtuse, then m1 = 120
3. Write the ~p ~q in words (inverse)
4. Write the ~q ~p in words (contrapositive)
Symbolic Notation - Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Let p be “m1 = 120”, and let q be “1 is obtuse”
1. Write the p q in words (conditional) If m1 = 120, then 1 is obtuse
2. Write the q p in words (converse) If 1 is obtuse, then m1 = 120
3. Write the ~p ~q in words (inverse) If m1 ≠ 120, then 1 is not obtuse
4. Write the ~q ~p in words (contrapositive)
Symbolic Notation - Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Let p be “m1 = 120”, and let q be “1 is obtuse”
1. Write the p q in words (conditional) If m1 = 120, then 1 is obtuse
2. Write the q p in words (converse) If 1 is obtuse, then m1 = 120
3. Write the ~p ~q in words (inverse) If m1 ≠ 120, then 1 is not obtuse
4. Write the ~q ~p in words (contrapositive) If 1 is not obtuse, then m1 ≠ 120
Symbolic Notation - Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Let p be “m1 = 90”, and let q be “1 is a right angle”
1. Write the p q in words (biconditional)
Symbolic Notation - Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Let p be “m1 = 90”, and let q be “1 is a right angle”
1. Write the p q in words (biconditional)m1 = 90 if and only if 1 is a right angle
Truth Table - Implication
ERHS Math Geometry
Mr. Chin-Sung Lin
Implication: p q
The statement “p implies q” means that if p is true, then q must be also true
Truth Table - Implication
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Conditional
Truth Table - Conditional
ERHS Math Geometry
Mr. Chin-Sung Lin
P: you get >90 in all tests q: you pass the class
pq: 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
Conditional
Truth Table - Converse
ERHS Math Geometry
Mr. Chin-Sung Lin
P: you get >90 in all tests q: you pass the class
qp: 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
Converse
Truth Table - Inverse
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Inverse
Truth Table - Contrapositive
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Contrapositive
Truth Table - Summary
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Truth Table - Summary
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Truth Table - Equivalent Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
The conditional and the contrapositive are equivalent statements (logical equivalents)
pq 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
Truth Table - Equivalent Statements
ERHS Math Geometry
Mr. Chin-Sung Lin
The converse and the inverse are equivalent statements (logical equivalents)
qp 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
Equivalent Statements : Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Write the logical equivalent for the statement “If a polygon is a triangle, then it has three sides.”
Equivalent Statements : Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Equivalent Statements : Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
Write the logical equivalent for the statement “If two nonintersecting lines are not coplanar, then they are skew line.”
Equivalent Statements : Exercise
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
A biconditional is true when two statements are both true or both false
When two statements have different truth values, the biconditional is false
Truth Table - Biconditional
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Applications of Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Applications of Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Applications of Biconditionals
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Laws of Logic
ERHS Math Geometry
Mr. Chin-Sung Lin
Laws of Logic
ERHS Math Geometry
Mr. Chin-Sung Lin
The thought patterns used to combine the known facts in order to establish the truth of related facts and draw conclusions
Laws of Logic - Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
If a conditional (pq) 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
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
If two segment have the same length, then they are congruent
You know that AB = CD
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
Johnson watches TV every Thursday and Saturday night
Today is Thursday
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
Johnson watches TV every Thursday and Saturday night
Today is Thursday
So, Johnson will watch TV tonight
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
All men will die
Mr. Lin is a man
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
All men will die
Mr. Lin is a man
So, Mr. Lin will die
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
All human will die
Mr. Lin does not die
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
All human will die
Mr. Lin does not die
So, Mr. Lin is not human
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
Vertical angles are congruent
A and C are vertical angles
Law of Detachment
ERHS Math Geometry
Mr. Chin-Sung Lin
Vertical angles are congruent
A and C are vertical angles
then, A C
Laws of Logic - Law of Disjunctive Inference
ERHS Math Geometry
Mr. Chin-Sung Lin
Law of Disjunctive Inference
When a disjunction is true and one of the disjuncts is false, then the other disjunct must be true
Law of Disjunctive Inference
ERHS Math Geometry
Mr. Chin-Sung Lin
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 p V q
T T T
T F T
F T T
F F F
Law of Disjunctive Inference
ERHS Math Geometry
Mr. Chin-Sung Lin
I walk to school or I take bus to school
I do not walk to school
Law of Disjunctive Inference
ERHS Math Geometry
Mr. Chin-Sung Lin
I walk to school or I take bus to school
I do not walk to school
So, I take bus to school
Law of Disjunctive Inference
ERHS Math Geometry
Mr. Chin-Sung Lin
Johnson watches TV every Thursday or Saturday
Johnson does not watche TV this Thursday
Law of Disjunctive Inference
ERHS Math Geometry
Mr. Chin-Sung Lin
Johnson watches TV every Thursday or Saturday
Johnson does not watch TV this Thursday
So, Johnson will watch TV this Saturday
Laws of Logic - Law of Syllogism
ERHS Math Geometry
Mr. Chin-Sung Lin
Law of Syllogism - Chain Rule
If hypothesis p, then conclusion qIf hypothesis q, then conclusion r
If hypothesis p, then conclusion r
If these statements
are true
then this statement
is true
Law of Syllogism
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Law of Syllogism
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Law of Syllogism
ERHS Math Geometry
Mr. Chin-Sung Lin
If x2 > 25, then x2 > 20If x > 5, then x2 > 25
Law of Syllogism
ERHS Math Geometry
Mr. Chin-Sung Lin
If x2 > 25, then x2 > 20If x > 5, then x2 > 25
If x > 5, then x2 > 20
The order of the statement doesn’t affect the application of the law of syllogism
Law of Syllogism
ERHS Math Geometry
Mr. Chin-Sung Lin
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
Drawing Conclusions
ERHS Math Geometry
Mr. Chin-Sung Lin
Drawing conclusions
ERHS Math Geometry
Mr. Chin-Sung Lin
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 sing2. Rachel will join the choir or Rachel will play
basketball3. Rachel does not like to sing
Drawing conclusions
ERHS Math Geometry
Mr. Chin-Sung Lin
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 sing2. Rachel will join the choir or Rachel will play
basketball3. 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”
Drawing conclusions
ERHS Math Geometry
Mr. Chin-Sung Lin
Original statements1. If Rachel joins the choir then Rachel likes to sing2. Rachel will join the choir or Rachel will play
basketball3. Rachel does not like to sing
Convert to symbolic form1. c s2. c V b3. ~s
Drawing conclusions
ERHS Math Geometry
Mr. Chin-Sung Lin
Symbolic form1. c s2. c V b3. ~s
Draw conclusions1. 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)
Drawing conclusions
ERHS Math Geometry
Mr. Chin-Sung Lin
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 sing2. Rachel will join the choir or Rachel will play
basketball3. Rachel does not like to sing
Conclusionb is true, so, “Rachel will play basketball“
Q & A
ERHS Math Geometry
Mr. Chin-Sung Lin
The End
ERHS Math Geometry
Mr. Chin-Sung Lin