Logic Eleanor Roosevelt High School Geometry Mr ChinSung

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

ERHS Math Geometry Sentences, Statements, and Truth Values Mr. Chin-Sung Lin

ERHS Math Geometry Logic is the science of reasoning The principles of logic allow us to determine if a statement is true, false, or uncertain on the basis of the truth of related statements Mr. Chin-Sung Lin

ERHS Math Geometry Sentences and Truth Values When we can determine that a statement is true or that it is false, that statement is said to have a truth value Statements with known truth values can be combined by the laws of logic to determine the truth value of other statements Mr. Chin-Sung Lin

ERHS Math Geometry Mathematical Sentences Simple declarative statements that state a fact, and that fact can be true or false • Parallel lines are coplanar TRUE • Straight angle is 180 o TRUE • x + (-x) = 1 FALSE • Obtuse triangle has 2 obtuse angles FALSE Mr. Chin-Sung Lin

ERHS Math Geometry Nonmathematical Sentences that do not state a fact, such as questions, commands, phrases, or exclamations • Is geometry hard? Question • Straight angle is 180 o Command • All the isosceles triangles Phrase • Wow! Exclamation Mr. Chin-Sung Lin

ERHS Math Geometry Nonmathematical Sentences We will not discuss sentences that are true for some persons and false for others • I love winter • Basket ball is the best sport • Triangle is the most beautiful geometric shape Mr. Chin-Sung Lin

ERHS Math Geometry Open Sentences that contain a variable The truth vale of the open sentence depends on the value of the variable • AB = 20 Variable: AB • 2 x + 3 = 15 Variable: x • He got 95 in geometry test Variable: he Mr. Chin-Sung Lin

ERHS Math Geometry Open Sentences Domain or Replacement Set The set of all elements that are possible replacements for the variable Solution Set or Truth Set The element(s) from the domain that make the open sentence true Mr. Chin-Sung Lin

ERHS Math Geometry Solution Set or Truth Set Example: Open sentence: Variable: Domain: Solution set: x + 5 = 10 x all real numbers 5 Mr. Chin-Sung Lin

ERHS Math Geometry Solution Set or Truth Set Example: Open sentence: Variable: Domain: Solution set: x (1/x) = 10 x all real numbers Φ, { }, or empty set Mr. Chin-Sung Lin

ERHS Math Geometry Exercise Identify each of the following sentences as true, false, open, or nonmathematical • Add A and B NONMATH • Congruent lines are always parallel FALSE • 3(x – 2) = 2(x – 3) + x TRUE • y – 6 = 2 y + 7 OPEN • Is ΔABC an equilateral triangle? NONMATH • Distance between 2 points is positive TRUE Mr. Chin-Sung Lin

ERHS Math Geometry Exercise Use the replacement set {3, 3. 14, √ 3, 1/3, 3π} to find the truth set of the open sentence “It is a rational number. ” Truth Set: {3, 3. 14, 1/3} Mr. Chin-Sung Lin

ERHS Math Geometry Statements and Symbols A sentence that has a truth value is called a statement or a closed sentence Truth value can be true [T] or false [F] In a statement, there are no variables Mr. Chin-Sung Lin

ERHS Math Geometry Negations The negation of a statement always has the opposite truth value of the original statement and is usually formed by adding the word not to the given statement • Statement Right angle is 90 o TRUE • Negation FALSE Right angle is not 90 o • Statement Triangle has 4 sides • Negation Triangle does not have 4 sides FALSE TRUE Mr. Chin-Sung Lin

ERHS Math Geometry Logic Symbols The basic element of logic is a simple declarative sentence We represent this element by a lowercase letter (p, q, r, and s are the most common) • Statement Right angle is 90 o TRUE • Negation FALSE Right angle is not 90 o • Statement Triangle has 4 sides • Negation Triangle does not have 4 sides FALSE TRUE Mr. Chin-Sung Lin

ERHS Math Geometry Logic Symbols The basic element of logic is a simple declarative sentence We represent this element by a lowercase letter (p, q, r, and s are the most common) Mr. Chin-Sung Lin

ERHS Math Geometry Logic Symbols For example, Statement p represents Right angle is 90 o Negation ~p represents Right angle is not 90 o ~p is read “not p” Mr. Chin-Sung Lin

ERHS Math Geometry Logic Symbols Symbol P Statement Truth value There are 3 sides in a triangle T There are not 3 sides in a triangle F q 2 x + 3 = 2 x F ~q 2 x + 3 ≠ 2 x T r NYC is a city T NYC is not a city F ~p ~r Mr. Chin-Sung Lin

ERHS Math Geometry Logic Symbols Symbol r ~r ~(~r) Statement Truth value NYC is a city T NYC is not a city F It is not true that NYC is not a city T T ~(~r) always has the same truth value as r ~r ~(~r) NYC is not a city F NYC is a city T Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table The relationship between a statement p and its negation ~p can be summarized in a truth table A statement p and its negation ~p have opposite truth values p ~p T F F T Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions Mr. Chin-Sung Lin

ERHS Math Geometry Compound Sentences / Statements Mathematical sentences formed by connectives such as and or Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions A compound statement formed by combining two simple statements using the word and Each of the simple statements is called a conjunct Statement: Conjunction Symbols: p, q p and q p^q Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions Example: p: A week has 7 days (T) q: A day has 24 hours (T) p^q: A week has 7 days and a day has 24 hours (T) Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions A conjunction is true when both statements are true When one or both statements are false, the conjunction is false Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions Example: p: A week has 7 days (T) q: A day does not have 24 hours (F) p^q: A week has 7 days and a day does not have 24 hours (F) Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions Tree Diagram q is true p ^ q is true q is false p ^ q is false q is true p ^ q is false p is true p is false Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions Truth Table p q p^q T T F F F T F F Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions Example: p: 3 is an odd number (T) q: 4 is an even number (T) p^q: 3 is an odd number and 4 is an even number (T) p q p^q T T T Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions A conjunction may contain a statement and a negation at the same time p q ~q p ^ ~q T T F F F F T F Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions Example: p: 3 is an odd number (T) q: 5 is an even number (F) p^~q: 3 is an odd number and 5 is not an even number (T) p q ~q p ^ ~q T F T T Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions A conjunction may contain a statement and a negation at the same time p q ~p ~p ^ q T T F F F F T T T F F T F Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions Example: p: 2 is an odd number (F) q: 4 is an even number (T) ~p^q: 2 is not an odd number and 4 is an even number (T) p q ~p ~p ^ q F T T T Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions A conjunction may contain two negations at the same time p q ~p ~q ~p ^ ~q T T F F F T T F F T T T Mr. Chin-Sung Lin

ERHS Math Geometry Conjunctions Example: p: 2 is an odd number (F) q: 5 is and even number (F) ~p^~q: 2 is not an odd number and 5 is not an even number (T) p q ~p ~q ~p ^ ~q F F T T T Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions A compound statement formed by combining two simple statements using the word or Each of the simple statements is called a disjunct Statement: Disjunction Symbols: p, q p or q p. Vq Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Example: p: A week has 7 days (T) q: A day has 20 hours (F) p. Vq: A week has 7 days or a day has 20 hours (T) Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions A disjunction is true when one or both statements are true When both statements are false, the disjunction is false Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Example: p: A week has 8 days (F) q: A day does not have 24 hours (F) p. Vq: A week has 8 days or a day does not have 24 hours (F) Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Tree Diagram q is true p V q is true q is false p V q is false p is true p is false Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Truth Table p q p. Vq T T F F F Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Example: p: 3 is an odd number (T) q: 5 is an even number (F) p. Vq: 3 is an odd number or 5 is an even number (T) p q p. Vq T F T Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions A disjunction may contain a statement and a negation at the same time p q ~q p V ~q T T F F F F T T Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Example: p: 3 is an odd number (T) q: 5 is an even number (F) p. V~q: 3 is an odd number or 5 is not an even number (T) p q ~q p V ~q T F T T Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions A disjunction may contain a statement and a negation at the same time p q ~p ~p V q T T F F F F T T T F F T T Mr. Chin-Sung Lin

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

ERHS Math Geometry Disjunctions A disjunction may contain two negations at the same time p q ~p ~q ~p V ~q T T F F F T T F T F F T T T Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Example: p: 2 is an odd number (F) q: 5 is an even number (F) ~p. V~q: 2 is not an odd number or 5 is not an even number (T) p q ~p ~q ~p V ~q F F T T T Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Use the following statements: Let k represent “Kurt plays baseball. ” Let a represent “Alicia plays baseball. ” Let n represent “Nathan plays soccer. ” Write each given sentence in symbolic form: a. Kurt or Alicia play baseball b. Kurt plays baseball or Nathan plays soccer Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Use the following statements: Let k represent “Kurt plays baseball. ” Let a represent “Alicia plays baseball. ” Let n represent “Nathan plays soccer. ” Write each given sentence in symbolic form: a. Kurt or Alicia play baseball (k V a) b. Kurt plays baseball or Nathan plays soccer (k V n) Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Use the following statements: Let k represent “Kurt plays baseball. ” Let a represent “Alicia plays baseball. ” Let n represent “Nathan plays soccer. ” Write each given sentence in symbolic form: a. Alicia plays baseball or Alicia does not play baseball b. It is not true that Kurt or Alicia play baseball Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Use the following statements: Let k represent “Kurt plays baseball. ” Let a represent “Alicia plays baseball. ” Let n represent “Nathan plays soccer. ” Write each given sentence in symbolic form: a. Alicia plays baseball or Alicia does not play baseball (a V ~a) b. It is not true that Kurt or Alicia play baseball (~(k V a)) Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Use the following statements: Let k represent “Kurt plays baseball. ” Let a represent “Alicia plays baseball. ” Let n represent “Nathan plays soccer. ” Write each given sentence in symbolic form: a. Either Kurt does not play baseball or Alicia does not play baseball b. It’s not the case that Alicia or Kurt play baseball Mr. Chin-Sung Lin

ERHS Math Geometry Disjunctions Use the following statements: Let k represent “Kurt plays baseball. ” Let a represent “Alicia plays baseball. ” Let n represent “Nathan plays soccer. ” Write each given sentence in symbolic form: a. Either Kurt does not play baseball or Alicia does not play baseball (~k V ~a) b. It’s not the case that Alicia or Kurt play baseball (~ (a V k)) Mr. Chin-Sung Lin

ERHS Math Geometry Inclusive OR vs. Exclusive OR When we use the word or to mean that one or both of the simple sentences are true, we call this the inclusive or When we use the word or to mean that one and only one of the simple sentences is true, we call this the exclusive or In the exclusive or, the disjunction p or q will be true when p is true, or when q is true, but not both Mr. Chin-Sung Lin

ERHS Math Geometry Exclusive OR Truth Table p q p⊕q T T F T F T T F F F Mr. Chin-Sung Lin

ERHS Math Geometry Example Find the solution set of each of the following if the domain is the set of positive integers less than 8 a. (x < 4) ∨ (x > 3) b. (x > 3) ∨ (x is odd) c. (x > 5) ∧ (x < 3) Mr. Chin-Sung Lin

ERHS Math Geometry Example Find the solution set of each of the following if the domain is the set of positive integers less than 8 a. (x < 4) ∨ (x > 3) {1, 2, 3, 4, 5, 6, 7} b. (x > 3) ∨ (x is odd) {1, 3, 4, 5, 6, 7} c. (x > 5) ∧ (x < 3) {} Mr. Chin-Sung Lin

ERHS Math Geometry Conditionals Mr. Chin-Sung Lin

ERHS Math Geometry Conditionals (or Implications) A compound statement formed by using the word if…. . then to combine two simple statements Statement: Conditional: Symbols: p, q if p then q p implies q p only if q p q Mr. Chin-Sung Lin

ERHS Math Geometry Conditionals Example: p: It is raining q: The street is wet p q: If it is raining then the road is wet q p: If the street is wet then it is raining * when we change the order of two statements in conditional, we may not have the same truth value as the original Mr. Chin-Sung Lin

ERHS Math Geometry Parts of a Conditional Statement A conditional statement is a logical statement that has two parts: a hypothesis (premise, antecedent) and a conclusion (consequent) Hypothesis Conclusion Mr. Chin-Sung Lin

ERHS Math Geometry Parts of a Conditional Statement A conditional statement is a logical statement that has two parts: a hypothesis (premise, antecedent) and a conclusion (consequent) Hypothesis Conclusion an assertion or a sentence that begins an argument Mr. Chin-Sung Lin

ERHS Math Geometry Parts of a Conditional Statement A conditional statement is a logical statement that has two parts: a hypothesis (premise, antecedent) and a conclusion (consequent) Hypothesis Conclusion the part of a sentence that closes an argument Mr. Chin-Sung Lin

ERHS Math Geometry Parts of a Conditional Statement When a conditional statement is in if-then form, the if part contains the hypothesis and then part contains the conclusion. IF Hypothesis THEN Conclusion Mr. Chin-Sung Lin

ERHS Math Geometry Parts of a Conditional Statement Example: If two angles form a linear pair, then these angles are supplementary IF ΔABC is equiangular Hypothesis THEN one of the angles is 60 o Conclusion Mr. Chin-Sung Lin

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

ERHS Math Geometry Truth Values for the Conditional p q Example Case 1: p: It is January (T) q: It is winter (T) p q: If it is January then it is winter (T) Mr. Chin-Sung Lin

ERHS Math Geometry Truth Values for the Conditional p q Example Case 2: p: It is January (T) q: It is winter (F) p q: If it is January then it is winter (F) Mr. Chin-Sung Lin

ERHS Math Geometry Truth Values for the Conditional p q Example Case 3: p: It is January (F) q: It is winter (T) p q: If it is January then it is winter (T) Mr. Chin-Sung Lin

ERHS Math Geometry Truth Values for the Conditional p q Example Case 4: p: It is January (F) q: It is winter (F) p q: If it is January then it is winter (T) Mr. Chin-Sung Lin

ERHS Math Geometry Truth Values for the Conditional p q A conditional is false when a true hypothesis leads to a false condition In all other cases, the conditional is true Mr. Chin-Sung Lin

ERHS Math Geometry Truth Values for the Conditional p q Tree Diagram q is true p q is true q is false p q is false q is true p q is true q is false p q is true p is false Mr. Chin-Sung Lin

ERHS Math Geometry Truth Values for the Conditional p q Truth Table p q T T F F F T T F F T Mr. Chin-Sung Lin

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

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

ERHS Math Geometry Rewrite a Statement in If-Then Form When I finish my homework, I will go to sleep Mr. Chin-Sung Lin

ERHS Math Geometry Rewrite a Statement in If-Then Form When I finish my homework, I will go to sleep If I finish my homework, then I will go to sleep Mr. Chin-Sung Lin

ERHS Math Geometry Rewrite a Statement in If-Then Form The homework is easy if I pay attention in class Mr. Chin-Sung Lin

ERHS Math Geometry Rewrite a Statement in If-Then Form The homework is easy if I pay attention in class If I pay attention in class, then the homework is easy Mr. Chin-Sung Lin

ERHS Math Geometry Rewrite a Statement in If-Then Form Linear pairs are supplementary Mr. Chin-Sung Lin

ERHS Math Geometry Rewrite a Statement in If-Then Form Linear pairs are supplementary If two angles form a linear pair, then these angles are supplementary Mr. Chin-Sung Lin

ERHS Math Geometry Rewrite a Statement in If-Then Form Two right angles are congruent Mr. Chin-Sung Lin

ERHS Math Geometry Rewrite a Statement in If-Then Form Two right angles are congruent If two angles are right angles, then these angles are congruent Mr. Chin-Sung Lin

ERHS Math Geometry Rewrite a Statement in If-Then Form Vertical angles are congruent Mr. Chin-Sung Lin

ERHS Math Geometry Rewrite a Statement in If-Then Form Vertical angles are congruent If two angles are vertical angles, then these angles are congruent Mr. Chin-Sung Lin

ERHS Math Geometry Verify a Conditional Statement A conditional statement can be true or false To show that a conditional statement is true, you need to prove that the conclusion is true every time the hypothesis is true To show that a conditional statement is false, you need to give only one counterexample Mr. Chin-Sung Lin

ERHS Math Geometry Verify a Conditional Statement Example: If two angles are vertical angles, then these angles are congruent During the prove process, you can not assume that these two angles are of certain degrees, the proof needs to cover all the possible vertical angle pairs Mr. Chin-Sung Lin

ERHS Math Geometry Conditionals, Inverses, Contrapositives & Biconditionals Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement 2. Converse 3. Inverse 4. Contrapositive 5. Biconditionsls Mr. Chin-Sung Lin

ERHS Math Geometry Converse To write the converse of a conditional statement, exchange the hypothesis and conclusion Statement: If m 1 = 120, then 1 is obtuse Converse: If 1 is obtuse, then m 1 = 120 Mr. Chin-Sung Lin

ERHS Math Geometry Inverse To write the inverse of a conditional statement, negate both the hypothesis and conclusion Statement: If m 1 = 120, then 1 is obtuse Inverse: If m 1 ≠ 120, then 1 is not obtuse Mr. Chin-Sung Lin

ERHS Math Geometry Contrapositive To write the contrapositive of a conditional statement, first write the converse, and then negate both the hypothesis and conclusion Statement: If m 1 = 120, then 1 is obtuse Contrapositive: If 1 is not obtuse, then m 1 ≠ 120 Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement If m 1 = 120, then 1 is obtuse 2. Converse If 1 is obtuse, then m 1 = 120 3. Inverse If m 1 ≠ 120, then 1 is not obtuse 4. Contrapositive If 1 is not obtuse, then m 1 ≠ 120 Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement If you are a basketball player, then you are an athlete 2. Converse 3. Inverse 4. Contrapositive Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement If you are a basketball player, then you are an athlete 2. Converse If you are an athlete, then you are a basketball player 3. Inverse 4. Contrapositive Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement If you are a basketball player, then you are an athlete 2. Converse If you are an athlete, then you are a basketball player 3. Inverse If you are not a basketball player, then you are not an athlete 4. Contrapositive Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement If you are a basketball player, then you are an athlete 2. Converse If you are an athlete, then you are a basketball player 3. Inverse If you are not a basketball player, then you are not an athlete 4. Contrapositive If you are not an athlete, then you are not a basketball player Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement (TRUE) If you are a basketball player, then you are an athlete 2. Converse If you are an athlete, then you are a basketball player 3. Inverse If you are not a basketball player, then you are not an athlete 4. Contrapositive If you are not an athlete, then you are not a basketball player Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement (TRUE) If you are a basketball player, then you are an athlete 2. Converse (FALSE) If you are an athlete, then you are a basketball player 3. Inverse If you are not a basketball player, then you are not an athlete 4. Contrapositive If you are not an athlete, then you are not a basketball player Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement (TRUE) If you are a basketball player, then you are an athlete 2. Converse (FALSE) If you are an athlete, then you are a basketball player 3. Inverse (FALSE) If you are not a basketball player, then you are not an athlete 4. Contrapositive If you are not an athlete, then you are not a basketball player Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement (TRUE) If you are a basketball player, then you are an athlete 2. Converse (FALSE) If you are an athlete, then you are a basketball player 3. Inverse (FALSE) If you are not a basketball player, then you are not an athlete 4. Contrapositive (TRUE) If you are not an athlete, then you are not a basketball player Mr. Chin-Sung Lin

ERHS Math Geometry Related Conditional Statements 1. Conditional Statement (TRUE) If you are a basketball player, then you are an athlete 2. Converse (FALSE) If you are an athlete, then you are a basketball player 3. Inverse (FALSE) If you are not a basketball player, then you are not an athlete 4. Contrapositive (TRUE) If you are not an athlete, then you are not a basketball player Mr. Chin-Sung Lin

ERHS Math Geometry Biconditional Statements When a conditional statement and its converse are both true, you can write them as a single biconditional statement A biconditional is the conjunction of a conditional and its converse A biconditional statement is a statement that contains the phrase “if and only if” Mr. Chin-Sung Lin

ERHS Math Geometry Biconditional Statements Statement If two lines intersect to form a right angle, then they are perpendicular Converse If two lines are perpendicular, then they intersect to form a right angle Bidirectional statement Two lines are perpendicular if and only if they intersect to form a right angle Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation Conditional statements can be written using symbolic notation: Letters (e. g. p) “statements” Arrow ( ) “implies” connects the hypothesis and conclusion Negation (~) “not” negates a statement as ~p Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Conditional Statement If two lines intersect to form a right angle, then they are perpendicular Let p be “two lines intersect to form a right angle” Let q be “they are perpendicular” If p, then q p q Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Converse Conditional Statement If two lines intersect to form a right angle, then they are perpendicular If p, then q p q Converse If two lines are perpendicular, then they intersect to form a right angle If q, then p q p Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Inverse Conditional Statement If two lines intersect to form a right angle, then they are perpendicular If p, then q p q Inverse If two lines intersect not to form a right angle, then they are not perpendicular If not p, then not q ~p ~q Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Contrapositive Conditional Statement If two lines intersect to form a right angle, then they are perpendicular If p, then q p q Contrapositive If two lines are not perpendicular, then they intersect not to form a right angle If not q, then not p ~q ~p Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Biconditional Conditional Statement If two lines intersect to form a right angle, then they are perpendicular If p, then q p q Biconditional Two lines intersect to form a right angle if and only if they are perpendicular p if and only if q p q Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Summary Conditional Statement If p, then q p q Converse If q, then p q p If not p, then not q ~p ~q If not q, then not p ~q ~p Inverse Contrapositive Biconditional p if and only if q p q Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Exercise Let p be “m 1 = 120”, and let q be “ 1 is obtuse” 1. Write the p q in words (conditional) 2. Write the q p in words (converse) 3. Write the ~p ~q in words (inverse) 4. Write the ~q ~p in words (contrapositive) Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Exercise Let p be “m 1 = 120”, and let q be “ 1 is obtuse” 1. Write the p q in words (conditional) If m 1 = 120, then 1 is obtuse 2. Write the q p in words (converse) 3. Write the ~p ~q in words (inverse) 4. Write the ~q ~p in words (contrapositive) Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Exercise Let p be “m 1 = 120”, and let q be “ 1 is obtuse” 1. Write the p q in words (conditional) If m 1 = 120, then 1 is obtuse 2. Write the q p in words (converse) If 1 is obtuse, then m 1 = 120 3. Write the ~p ~q in words (inverse) 4. Write the ~q ~p in words (contrapositive) Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Exercise Let p be “m 1 = 120”, and let q be “ 1 is obtuse” 1. Write the p q in words (conditional) If m 1 = 120, then 1 is obtuse 2. Write the q p in words (converse) If 1 is obtuse, then m 1 = 120 3. Write the ~p ~q in words (inverse) If m 1 ≠ 120, then 1 is not obtuse 4. Write the ~q ~p in words (contrapositive) Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Exercise Let p be “m 1 = 120”, and let q be “ 1 is obtuse” 1. Write the p q in words (conditional) If m 1 = 120, then 1 is obtuse 2. Write the q p in words (converse) If 1 is obtuse, then m 1 = 120 3. Write the ~p ~q in words (inverse) If m 1 ≠ 120, then 1 is not obtuse 4. Write the ~q ~p in words (contrapositive) If 1 is not obtuse, then m 1 ≠ 120 Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Exercise Let p be “m 1 = 90”, and let q be “ 1 is a right angle” 1. Write the p q in words (biconditional) Mr. Chin-Sung Lin

ERHS Math Geometry Symbolic Notation - Exercise Let p be “m 1 = 90”, and let q be “ 1 is a right angle” 1. Write the p q in words (biconditional) m 1 = 90 if and only if 1 is a right angle Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Implication: p q The statement “p implies q” means that if p is true, then q must be also true Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Implication Conditional For hypothesis p and conclusion q: The condition p q is only false when a true hypothesis produce a false conclusion p q T T F F F T T F F T Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Conditional P: you get >90 in all tests q: you pass the class p q: If you get >90 in all tests then you pass the class p q T T F F F T T F F T Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Converse P: you get >90 in all tests q: you pass the class q p: If you pass the class then you get >90 in all tests p q q p T T F T F F F T Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Inverse P: you get >90 in all tests q: you pass the class ~p ~q: If you don’t get >90 in all tests then you don’t pass the class p q ~p ~q T T F T F F F T Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Contrapositive P: you get >90 in all tests q: you pass the class ~q ~p: If you don’t pass the class then you don’t get >90 in all tests p q ~q ~p T T F F F T T F F T Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Summary p q q p ~p ~q ~q ~p T T T T F F T T Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Summary p q q p ~p ~q ~q ~p T T T T F F T T Equivalent Statements Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Equivalent Statements The conditional and the contrapositive are equivalent statements (logical equivalents) p q If you get >90 in all tests, then you pass the class ~q ~p If you don’t pass the class, then you don’t get >90 in all tests Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Equivalent Statements The converse and the inverse are equivalent statements (logical equivalents) q p If you pass the class, then you get >90 in all tests ~p ~q If you don’t get >90 in all tests, then you don’t pass the class Mr. Chin-Sung Lin

ERHS Math Geometry Equivalent Statements : Exercise Write the logical equivalent for the statement “If a polygon is a triangle, then it has three sides. ” Mr. Chin-Sung Lin

ERHS Math Geometry Equivalent Statements : Exercise Write the logical equivalent for the statement “If a polygon is a triangle, then it has three sides. ” If a polygon does not have three sides, then it is not a triangle Mr. Chin-Sung Lin

ERHS Math Geometry Equivalent Statements : Exercise Write the logical equivalent for the statement “If two nonintersecting lines are not coplanar, then they are skew line. ” Mr. Chin-Sung Lin

ERHS Math Geometry Equivalent Statements : Exercise Write the logical equivalent for the statement “If two nonintersecting lines are not coplanar, then they are skew line. ” If two nonintersecting lines are not skew lines, then they are coplanar Mr. Chin-Sung Lin

ERHS Math Geometry Biconditionals A biconditional is true when two statements are both true or both false When two statements have different truth values, the biconditional is false Mr. Chin-Sung Lin

ERHS Math Geometry Truth Table - Biconditional p q q p (p q) ^ (q p) p q T T T T F F F T T Mr. Chin-Sung Lin

ERHS Math Geometry Applications of Biconditionals Definitions are true biconditionals • Right angles are angles with measure of 90 • Angles with measure of 90 are right angles • Congruent segments are segments with the same measure • Segments with the same measure are congruent segments Mr. Chin-Sung Lin

ERHS Math Geometry Applications of Biconditionals are used to solve equations • If x + 3 = 5, then x = 2 • If x = 2, then x + 3 = 5 * The solution of an equation is a series of biconditionals Mr. Chin-Sung Lin

ERHS Math Geometry Applications of Biconditionals state logical equivalents • ~(p ^ q) � (~p V ~q) p q ~p ~q p^q ~(p ^ q) ~p V ~q T T F F T T F F T T Mr. Chin-Sung Lin

ERHS Math Geometry Laws of Logic Mr. Chin-Sung Lin

ERHS Math Geometry Laws of Logic The thought patterns used to combine the known facts in order to establish the truth of related facts and draw conclusions Mr. Chin-Sung Lin

ERHS Math Geometry Laws of Logic - Law of Detachment - Direct Argument A valid argument uses a series of statements called premises that have known truth values to arrive at a conclusion If the hypothesis of a true conditional statement is true, then the conclusion is also true Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment If a conditional (p q) is true and the hypothesis (p) is true, then the conclusion (q) is true p q T T F F F T T F F T Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment If two segment have the same length, then they are congruent You know that AB = CD Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment If two segment have the same length, then they are congruent You know that AB = CD Since AB = CD satisfies the hypothesis of a true conditional statement, the conclusion is also true. So, AB CD Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment Johnson watches TV every Thursday and Saturday night Today is Thursday Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment Johnson watches TV every Thursday and Saturday night Today is Thursday So, Johnson will watch TV tonight Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment All men will die Mr. Lin is a man Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment All men will die Mr. Lin is a man So, Mr. Lin will die Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment All human will die Mr. Lin does not die Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment All human will die Mr. Lin does not die So, Mr. Lin is not human Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment Vertical angles are congruent A and C are vertical angles Mr. Chin-Sung Lin

ERHS Math Geometry Law of Detachment Vertical angles are congruent A and C are vertical angles then, A C Mr. Chin-Sung Lin

ERHS Math Geometry Laws of Logic - Law of Disjunctive Inference When a disjunction is true and one of the disjuncts is false, then the other disjunct must be true Mr. Chin-Sung Lin

ERHS Math Geometry Law of Disjunctive Inference If a disjunction (p. Vq) is true and the disjunct (p) is false, then the other disjunct (q) is true If a disjunction (p. Vq) is true and the disjunct (q) is false, then the other disjunct (p) is true p q p. Vq T T F F F Mr. Chin-Sung Lin

ERHS Math Geometry Law of Disjunctive Inference I walk to school or I take bus to school I do not walk to school Mr. Chin-Sung Lin

ERHS Math Geometry Law of Disjunctive Inference I walk to school or I take bus to school I do not walk to school So, I take bus to school Mr. Chin-Sung Lin

ERHS Math Geometry Law of Disjunctive Inference Johnson watches TV every Thursday or Saturday Johnson does not watche TV this Thursday Mr. Chin-Sung Lin

ERHS Math Geometry Law of Disjunctive Inference Johnson watches TV every Thursday or Saturday Johnson does not watch� TV this Thursday So, Johnson will watch TV this Saturday Mr. Chin-Sung Lin

ERHS Math Geometry Laws of Logic - Law of Syllogism - Chain Rule If hypothesis p, then conclusion q If hypothesis q, then conclusion r If these statements are true If hypothesis p, then conclusion r then this statement is true Mr. Chin-Sung Lin

ERHS Math Geometry Law of Syllogism If two angles are linear pair, then they are supplementary If two angles are supplementary, then the sum of the measure of these angles are equal to 180 Mr. Chin-Sung Lin

ERHS Math Geometry Law of Syllogism If two angles are linear pair, then they are supplementary If two angles are supplementary, then the sum of the measure of these angles are equal to 180 If two angles are linear pair, then the sum of the measure of these angles are equal to 180 Mr. Chin-Sung Lin

ERHS Math Geometry Law of Syllogism If x 2 > 25, then x 2 > 20 If x > 5, then x 2 > 25 Mr. Chin-Sung Lin

ERHS Math Geometry Law of Syllogism x 2 If > 25, then > 20 If x > 5, then x 2 > 25 The order of the statement doesn’t affect the application of the law of syllogism If x > 5, then x 2 > 20 Mr. Chin-Sung Lin

ERHS Math Geometry Law of Syllogism If two triangles are congruent, then their corresponding sides are congruent If two triangles are congruent, then their corresponding angles are congruent Neither statement’s conclusion is the same as other statement’s hypothesis. So, you cannot use law of syllogism to write another conditional statement Mr. Chin-Sung Lin

ERHS Math Geometry Drawing Conclusions Mr. Chin-Sung Lin

ERHS Math Geometry Drawing conclusions The three statements given below are each true. What conclusion can be found to be true? 1. If Rachel joins the choir then Rachel likes to sing 2. Rachel will join the choir or Rachel will play basketball 3. Rachel does not like to sing Mr. Chin-Sung Lin

ERHS Math Geometry Drawing conclusions The three statements given below are each true. What conclusion can be found to be true? 1. If Rachel joins the choir then Rachel likes to sing 2. Rachel will join the choir or Rachel will play basketball 3. Rachel does not like to sing Let c represent “Rachel joins the choir” s represent “Rachel likes to sing” b represent “Rachel will play basketball” Mr. Chin-Sung Lin

ERHS Math Geometry Drawing conclusions Original statements 1. If Rachel joins the choir then Rachel likes to sing 2. Rachel will join the choir or Rachel will play basketball 3. Rachel does not like to sing Convert to symbolic form 1. c s 2. c V b 3. ~s Mr. Chin-Sung Lin

ERHS Math Geometry Drawing conclusions Symbolic form 1. c s 2. c V b 3. ~s Draw conclusions 1. c s is true, so ~s ~c is true (contrapositive) 2. ~s is true, so ~c is true (law of detachment) 3. ~c is true, so c is false (negation) 4. c V b is true and c is false, so, b is true (law of disjunctive inference) Mr. Chin-Sung Lin

ERHS Math Geometry Drawing conclusions The three statements given below are each true. What conclusion can be found to be true? 1. If Rachel joins the choir then Rachel likes to sing 2. Rachel will join the choir or Rachel will play basketball 3. Rachel does not like to sing Conclusion b is true, so, “Rachel will play basketball“ Mr. Chin-Sung Lin

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

ERHS Math Geometry The End Mr. Chin-Sung Lin