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 180 o x + (-x) = 1 Obtuse triangle has 2 obtuse angles TRUE 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 180 o 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 variable The 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 = 10 Variable:x Domain:all real numbers Solution set:5

Solution Set or Truth Set ERHS Math Geometry Mr. Chin-Sung Lin Example: Open sentence: x (1/x) = 10 Variable:x Domain:all real numbers Solution 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, 3 π } 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 StatementRight angle is 90 o NegationRight angle is not 90 o TRUE FALSE StatementTriangle has 4 sides NegationTriangle 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) StatementRight angle is 90 o NegationRight angle is not 90 o TRUE FALSE StatementTriangle has 4 sides NegationTriangle 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 90 o Negation ~p represents Right angle is not 90 o ~p is read “not p”

Logic Symbols ERHS Math Geometry Mr. Chin-Sung Lin SymbolStatementTruth value P There are 3 sides in a triangle T ~p There are not 3 sides in a triangle F q 2x + 3 = 2xF ~q 2x + 3 ≠ 2xT r NYC is a cityT ~r NYC is not a cityF

Logic Symbols ERHS Math Geometry Mr. Chin-Sung Lin SymbolStatementTruth value r NYC is a cityT ~r NYC is not a cityF ~(~r) It is not true that NYC is not a cityT T ~(~r) always has the same truth value as r ~r NYC is not a cityF ~(~r) NYC is a cityT

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 TF FT

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 Tree Diagram

Conjunctions ERHS Math Geometry Mr. Chin-Sung Lin Truth Table pqp ^ q TTT TFF FTF FFF

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) pqp ^ q TTT

Conjunctions ERHS Math Geometry Mr. Chin-Sung Lin A conjunction may contain a statement and a negation at the same time pq~qp ^ ~q TTFF TFTT FTFF FFTF

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) pq~qp ^ ~q TFTT

Conjunctions ERHS Math Geometry Mr. Chin-Sung Lin A conjunction may contain a statement and a negation at the same time pq~p~p ^ q TTFF TFFF FTTT FFTF

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) pq~p~p ^ q FTTT

Conjunctions ERHS Math Geometry Mr. Chin-Sung Lin A conjunction may contain two negations at the same time pq~p~q~p ^ ~q TTFFF TFFTF FTTFF FFTTT

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) pq~p~q~p ^ ~q FFTTT

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) p V q: 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) p V q: 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 false Tree Diagram

Disjunctions ERHS Math Geometry Mr. Chin-Sung Lin Truth Table pqp V q TTT TFT FTT FFF

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) pqp V q TFT

Disjunctions ERHS Math Geometry Mr. Chin-Sung Lin A disjunction may contain a statement and a negation at the same time pq~qp V ~q TTFT TFTT FTFF FFTT

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) pq~qp V ~q TFTT

Disjunctions ERHS Math Geometry Mr. Chin-Sung Lin A disjunction may contain a statement and a negation at the same time pq~p~p V q TTFT TFFF FTTT FFTT

Disjunctions ERHS Math Geometry Mr. Chin-Sung Lin Example: p: 2 is an odd number (F) q: 4 is an even number (T) ~p V q: 2 is not an odd number or 4 is an even number (T) pq~p~p V q FTTT

Disjunctions ERHS Math Geometry Mr. Chin-Sung Lin A disjunction may contain two negations at the same time pq~p~q~p V ~q TTFFF TFFTT FTTFT FFTTT

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) pq~p~q~p V ~q FFTTT

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 pq p ⊕ q TTF TFT FTT FFF

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 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 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 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

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) HypothesisConclusion

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) HypothesisConclusion 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) HypothesisConclusion 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. HypothesisConclusion IFTHEN

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 equiangular IFTHEN one of the angles is 60 o HypothesisConclusion

Parts of a Conditional Statement ERHS Math Geometry Mr. Chin-Sung Lin ΔABC is equiangular IFTHEN HypothesisConclusion one of the angles is 60 o ΔABC is equiangular IMPLIES THAT HypothesisConclusion one of the angles is 60 o ΔABC is equiangular ONLY IF HypothesisConclusion one of the angles is 60 o

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) p  q: 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) p  q: 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) p  q: 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) p  q: 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 Tree Diagram

Truth Values for the Conditional p  q ERHS Math Geometry Mr. Chin-Sung Lin Truth Table pqp  q TTT TFF FTT FFT

Conditionals ERHS Math Geometry Mr. Chin-Sung Lin Example: p: ☐ ABCD is a rectangle (F) q: AB // CD (T) p  q: If ☐ ABCD is a rectangle then AB // CD (?) pqp  q FT

Conditionals ERHS Math Geometry Mr. Chin-Sung Lin Example: p: ☐ ABCD is a rectangle (F) q: AB // CD (T) p  q: If ☐ ABCD is a rectangle then AB // CD (T) pqp  q FTT

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 m1 = 120, then 1 is obtuse Converse: If 1 is obtuse, then m1 = 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 m1 = 120, then 1 is obtuse Inverse: If m1 ≠ 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 m1 = 120, then 1 is obtuse Contrapositive: If 1 is not obtuse, then m1 ≠ 120

Related Conditional Statements ERHS Math Geometry Mr. Chin-Sung Lin 1. Conditional Statement If m1 = 120, then 1 is obtuse 2. Converse If 1 is obtuse, then m1 = Inverse If m1 ≠ 120, then 1 is not obtuse 4. Contrapositive If 1 is not obtuse, then m1 ≠ 120

Related Conditional Statements ERHS Math Geometry Mr. Chin-Sung Lin 1. Conditional Statement If 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 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

Related Conditional Statements ERHS Math Geometry Mr. Chin-Sung Lin 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

Related Conditional Statements ERHS Math Geometry Mr. Chin-Sung Lin 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

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 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

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 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

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 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

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 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 T wo 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 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

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 Converse If 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 Inverse If 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 Contrapositive If 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 Biconditional Two 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 Statement If p, then q p  q Converse If q, then p q  p Inverse If not p, then not q ~p  ~q Contrapositive If not q, then not p ~q  ~p Biconditional p if and only if q p  q

Symbolic Notation - Exercise ERHS Math Geometry Mr. Chin-Sung Lin 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)

Symbolic Notation - Exercise ERHS Math Geometry Mr. Chin-Sung Lin 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)

Symbolic Notation - Exercise ERHS Math Geometry Mr. Chin-Sung Lin 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 = 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 “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 = Write the ~p  ~q in words (inverse) If m1 ≠ 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 “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 = 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

Symbolic Notation - Exercise ERHS Math Geometry Mr. Chin-Sung Lin Let p be “m1 = 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 “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

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 pq p  q TTT TFF FTT FFT Conditional

Truth Table - Conditional ERHS Math Geometry Mr. Chin-Sung Lin 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 pq p  q TTT TFF FTT FFT Conditional

Truth Table - Converse ERHS Math Geometry Mr. Chin-Sung Lin 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 pq q  p TTT TFT FTF FFT 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 pq ~p  ~q TTT TFT FTF FFT 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 pq ~q  ~p TTT TFF FTT FFT Contrapositive

Truth Table - Summary ERHS Math Geometry Mr. Chin-Sung Lin pq p  qq  p~p  ~q~q  ~p TTTTTT TFFTTF FTTFFT FFTTTT

Truth Table - Summary ERHS Math Geometry Mr. Chin-Sung Lin pq p  qq  p~p  ~q~q  ~p TTTTTT TFFTTF FTTFFT FFTTTT Equivalent Statements

Truth Table - Equivalent Statements ERHS Math Geometry Mr. Chin-Sung Lin 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

Truth Table - Equivalent Statements ERHS Math Geometry Mr. Chin-Sung Lin 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

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 pq p  qq  p(p  q) ^ (q  p)p  q TTTTTT TFFTFF FTTFFF FFTTTT

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) pq ~p~qp ^ q~(p ^ q)~p V ~q TTFFTFF TFFTFTT FTTFFTT FFTTFTT

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 (pq) is true and the hypothesis (p) is true, then the conclusion (q) is true pq p  q TTT TFF FTT FFT

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 pq p V q TTT TFT FTT FFF

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 q If 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 x 2 > 25, then x 2 > 20 If x > 5, then x 2 > 25

Law of Syllogism ERHS Math Geometry Mr. Chin-Sung Lin If x 2 > 25, then x 2 > 20 If x > 5, then x 2 > 25 If x > 5, then x 2 > 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 sing 2. Rachel will join the choir or Rachel will play basketball 3. 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 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”

Drawing conclusions ERHS Math Geometry Mr. Chin-Sung Lin 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

Drawing conclusions ERHS Math Geometry Mr. Chin-Sung Lin 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)

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 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“

Q & A ERHS Math Geometry Mr. Chin-Sung Lin

The End ERHS Math Geometry Mr. Chin-Sung Lin