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


ERHS Math Geometry

Mr. Chin-Sung Lin

Example:

Open sentence: x + 5 = 10Variable: xDomain: all real numbersSolution set: 5


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:


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:


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)


~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:


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:


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:



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:



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:



~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:



~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






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






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






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






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






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


ERHS Math Geometry

Mr. Chin-Sung Lin



an assertion or a sentence that begins an argument


ERHS Math Geometry

Mr. Chin-Sung Lin



the part of a sentence that closes an argument


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


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



ERHS Math Geometry

Mr. Chin-Sung Lin

ΔABC is equiangularIF THEN



ΔABC is equiangular IMPLIES THAT



ΔABC is equiangular ONLY IF



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)


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)


ERHS Math Geometry

Mr. Chin-Sung Lin

Example Case 3:

p: It is January (F)

q: It is winter (T)



ERHS Math Geometry

Mr. Chin-Sung Lin

Example Case 4:

p: It is January (F)

q: It is winter (F)



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


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


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


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


ERHS Math Geometry

Mr. Chin-Sung Lin

The homework is easy if I pay attention in class


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


ERHS Math Geometry

Mr. Chin-Sung Lin

Linear pairs are supplementary


ERHS Math Geometry

Mr. Chin-Sung Lin

Linear pairs are supplementary

If two angles form a linear pair, then these angles are supplementary


ERHS Math Geometry

Mr. Chin-Sung Lin

Two right angles are congruent


ERHS Math Geometry

Mr. Chin-Sung Lin

Two right angles are congruent

If two angles are right angles, then these angles are congruent


ERHS Math Geometry

Mr. Chin-Sung Lin

Vertical angles are congruent


ERHS Math Geometry

Mr. Chin-Sung Lin


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


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


Contrapositive:If 1 is not obtuse, then m1 ≠ 120


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


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


ERHS Math Geometry

Mr. Chin-Sung Lin


2. ConverseIf you are an athlete, then you are a basketball player

3. Inverse

4. Contrapositive


ERHS Math Geometry

Mr. Chin-Sung Lin



3. InverseIf you are not a basketball player, then you are not an athlete

4. Contrapositive


ERHS Math Geometry

Mr. Chin-Sung Lin




4. ContrapositiveIf you are not an athlete, then you are not a basketball player


ERHS Math Geometry

Mr. Chin-Sung Lin

1. Conditional Statement (TRUE)If you are a basketball player, then you are an athlete





ERHS Math Geometry

Mr. Chin-Sung Lin


2. Converse (FALSE)If you are an athlete, then you are a basketball player




ERHS Math Geometry

Mr. Chin-Sung Lin



3. Inverse (FALSE)If you are not a basketball player, then you are not an athlete



ERHS Math Geometry

Mr. Chin-Sung Lin



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



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



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



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)


ERHS Math Geometry

Mr. Chin-Sung Lin


1. Write the p q in words (conditional) If m1 = 120, then 1 is obtuse

2. Write the q p in words (converse)




ERHS Math Geometry

Mr. Chin-Sung Lin



2. Write the q p in words (converse) If 1 is obtuse, then m1 = 120




ERHS Math Geometry

Mr. Chin-Sung Lin




3. Write the ~p ~q in words (inverse) If m1 ≠ 120, then 1 is not obtuse



ERHS Math Geometry

Mr. Chin-Sung Lin




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


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)


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


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~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


~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.”


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


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.”


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


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


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


A and C are vertical angles

Law of Detachment

ERHS Math Geometry

Mr. Chin-Sung Lin


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


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


ERHS Math Geometry

Mr. Chin-Sung Lin

I walk to school or I take bus to school

I do not walk to school


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


ERHS Math Geometry

Mr. Chin-Sung Lin

Johnson watches TV every Thursday or Saturday

Johnson does not watche TV this Thursday


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




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


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




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

Logic Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin.

Documents

Transcript of Logic Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin.