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## Logic

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**Logic**Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin**ERHS Math Geometry**Sentences, Statements, and Truth Values Mr. Chin-Sung Lin**ERHS Math Geometry**Logic Logic is the science of reasoning The principles of logic allow us to determine if a statement is true, false, or uncertain on the basis of the truth of related statements Mr. Chin-Sung Lin**ERHS Math Geometry**Sentences and Truth Values When we can determine that a statement is true or that it is false, that statement is said to have a truth value Statements with known truth values can be combined by the laws of logic to determine the truth value of other statements Mr. Chin-Sung Lin**ERHS Math Geometry**Mathematical Sentences Simple declarative statements that state a fact, and that fact can be true or false • Parallel lines are coplanar TRUE • Straight angle is 180o TRUE • x + (-x) = 1 FALSE • Obtuse triangle has 2 obtuse angles FALSE Mr. Chin-Sung Lin**ERHS Math Geometry**Sentences that do not state a fact, such as questions, commands, phrases, or exclamations Nonmathematical Sentences • Is geometry hard? Question • Straight angle is 180o Command • All the isosceles triangles Phrase • Wow! Exclamation Mr. Chin-Sung Lin**ERHS Math Geometry**Nonmathematical Sentences We will not discuss sentences that are true for some persons and false for others • I love winter • Basket ball is the best sport • Triangle is the most beautiful geometric shape Mr. Chin-Sung Lin**ERHS Math Geometry**Open Sentences Sentences that contain a variable The truth vale of the open sentence depends on the value of the variable • AB = 20 Variable: AB • 2x + 3 = 15 Variable: x • He got 95 in geometry test Variable: he Mr. Chin-Sung Lin**ERHS Math Geometry**Domain or Replacement Set Open Sentences 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**Example: Open sentence: x + 5 = 10 Variable: x Domain: all real numbers Solution set: 5 Solution Set or Truth Set Mr. Chin-Sung Lin**ERHS Math Geometry**Example: Open sentence: x (1/x) = 10 Variable: x Domain: all real numbers Solution set: Φ, { }, or empty set Solution Set or Truth Set Mr. Chin-Sung Lin**ERHS Math Geometry**Identify each of the following sentences as true, false, open, or nonmathematical Exercise • Add A and B NONMATH • Congruent lines are always parallel FALSE • 3(x – 2) = 2(x – 3) + x TRUE • y – 6 = 2y + 7 OPEN • Is ΔABC an equilateral triangle? NONMATH • Distance between 2 points is positive TRUE Mr. Chin-Sung Lin**ERHS Math Geometry**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.” Exercise Truth Set: {3, 3.14, 1/3} Mr. Chin-Sung Lin**ERHS Math Geometry**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 Statements and Symbols Mr. Chin-Sung Lin**ERHS Math Geometry**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 Negations • Statement Right angle is 90o TRUE • Negation Right angle is not 90o FALSE • Statement Triangle has 4 sides FALSE • Negation Triangle does not have 4 sides TRUE Mr. Chin-Sung Lin**ERHS Math Geometry**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 • Statement Right angle is 90o TRUE • Negation Right angle is not 90o FALSE • Statement Triangle has 4 sides FALSE • Negation Triangle does not have 4 sides TRUE Mr. Chin-Sung Lin**ERHS Math Geometry**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 Mr. Chin-Sung Lin**ERHS Math Geometry**For example, Statement p represents Right angle is 90o Negation ~p represents Right angle is not 90o ~p is read “not p” Logic Symbols Mr. Chin-Sung Lin**ERHS Math Geometry**Symbol Statement Truth value P There are 3 sides in a triangle T Logic Symbols ~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 Mr. Chin-Sung Lin**ERHS Math Geometry**Symbol Statement Truth value r NYC is a city T Logic Symbols ~r NYC is not a city F ~(~r) It is not true that NYC is not a city T T ~(~r) always has the same truth value asr ~r NYC is not a city F ~(~r) NYC is a city T Mr. Chin-Sung Lin**ERHS Math Geometry**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 Truth Table Mr. Chin-Sung Lin**ERHS Math Geometry**Conjunctions Mr. Chin-Sung Lin**ERHS Math Geometry**Compound Sentences / Statements Mathematical sentences formed by connectives such as and and or Mr. Chin-Sung Lin**ERHS Math Geometry**Conjunctions A compound statement formed by combining two simple statements using the word and Each of the simple statements is called a conjunct Statement: p, q Conjunction p and q Symbols: 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**Tree Diagram Conjunctions q is true p ^ q is true p is true q is false p ^ q is false q is true p ^ q is false p is false q is false p ^ q is false Mr. Chin-Sung Lin**ERHS Math Geometry**Truth Table Conjunctions 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) Mr. Chin-Sung Lin**ERHS Math Geometry**A conjunction may contain a statement and a negation at the same time Conjunctions 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) Mr. Chin-Sung Lin**ERHS Math Geometry**A conjunction may contain a statement and a negation at the same time Conjunctions 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) Mr. Chin-Sung Lin**ERHS Math Geometry**A conjunction may contain two negations at the same time Conjunctions 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) 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: p, q Disjunction p or q Symbols: p V q Mr. Chin-Sung Lin**ERHS Math Geometry**Disjunctions Example: p: A week has 7 days (T) q: A day has 20 hours (F) pVq: A week has 7 days or a day has 20 hours (T) Mr. Chin-Sung Lin**ERHS Math Geometry**Disjunctions A disjunction is true when one or both statements are true When both statements are false, the disjunction is false Mr. Chin-Sung Lin**ERHS Math Geometry**Disjunctions Example: p: A week has 8 days (F) q: A day does not have 24 hours (F) pVq: A week has 8 days or a day does not have 24 hours (F) Mr. Chin-Sung Lin**ERHS Math Geometry**Tree Diagram Disjunctions q is true p V q is true p is true q is false p V q is true q is true p V q is true p is false q is false p V q is false Mr. Chin-Sung Lin**ERHS Math Geometry**Truth Table Disjunctions Mr. Chin-Sung Lin**ERHS Math Geometry**Disjunctions Example: p: 3 is an odd number (T) q: 5 is an even number (F) pVq: 3 is an odd number or 5 is an even number (T) Mr. Chin-Sung Lin**ERHS Math Geometry**A disjunction may contain a statement and a negation at the same time Disjunctions Mr. Chin-Sung Lin**ERHS Math Geometry**Disjunctions Example: p: 3 is an odd number (T) q: 5 is an even number (F) pV~q: 3 is an odd number or 5 is not an even number (T) Mr. Chin-Sung Lin**ERHS Math Geometry**A disjunction may contain a statement and a negation at the same time Disjunctions Mr. Chin-Sung Lin**ERHS Math Geometry**Disjunctions Example: p: 2 is an odd number (F) q: 4 is an even number (T) ~pVq: 2 is not an odd number or 4 is an even number (T) Mr. Chin-Sung Lin**ERHS Math Geometry**A disjunction may contain two negations at the same time Disjunctions Mr. Chin-Sung Lin**ERHS Math Geometry**Disjunctions Example: p: 2 is an odd number (F) q: 5 is an even number (F) ~pV~q: 2 is not an odd number or 5 is not an even number (T) Mr. Chin-Sung Lin