L OGIC

A conjecture is an educated guess that can be either true or false. A statement is a sentence that is either true or false but not both. Often represented by a letter such a p, q or r. The Truth value is the truth or falsity of a statement.

N EGATIONS The negation of a statement says it has the opposite meaning. symbol ~p or ~q read “not p or not q” Example: p: Suffolk is a city in Virginia. The negation would be: ~p: Suffolk is not a city in Virginia.

C OMPOUND S TATEMENT A compound statement is two or more statements that are joined together. Example: p: Richmond is a city in Virginia. q: Richmond is the capital of Virginia. p and q: Richmond is a city in Virginia and Richmond is the capital of Virginia.

C ONJUNCTION A conjunction is a compound statement formed by joining two or more statements with the word “AND”. Symbolic representation: “read” p and q * A conjunction is true IFF both statements are true.*

E XAMPLE : USE THE FOLLOWING STATEMENTS TO WRITE A COMPOUND STATEMENT FOR EACH CONJUNCTION THEN FIND IT ’ S TRUTH VALUE. p: One foot is 14 inches. q: September has 30 days r: A plane is defined by 3 non-collinear points. a) p and q One foot is 14 inches and September has 30 days. b) A plane is defined by 3 non-collinear points and one foot is 14 inches. c) September does not have 30 days and a plane is defined by 3 non-collinear points. d) One foot does not have 14 inches and a plane is defined by 3 mom-collinear points.

D ISJUNCTION A disjunction is a compound statement that joins two or more statements with the word “or”. Symbolic representation: “read” p or q *A disjunction is true if at least one of the statements are true.*

E XAMPLE : USE THE FOLLOWING STATEMENTS TO WRITE A COMPOUND STATEMENT FOR EACH DISJUNCTION THEN FIND IT ’ S TRUTH VALUE. p: is proper notation for “line AB” q: centimeters are metric units. r: 9 is prime number a) p or q is proper notation for “line AB” or centimeters are metric units. b) Centimeters are metric units or 9 is a prime number.

Lesson 2-1 Conditional Statements 9 C ONDITIONAL S TATEMENT Definiti on: A conditional statement is a statement that can be written in if-then form. “ If _____________, then ______________.” “if p, then q”. Symbolic Notation p → q

Lesson 2-1 Conditional Statements 10 C ONDITIONAL S TATEMENT Conditional Statements have two parts: The hypothesis is the part of a conditional statement that follows “if” (Usually denoted p.) The conclusion is the part of an if-then statement that follows “then” (Usually denoted q.) The hypothesis is the given information, or the condition. The conclusion is the result of the given information.

E XAMPLE Write the statement “ An angle of 40° is acute.” Hypothesis – An angle of 40° Represented by : p Conclusion – is Acute Represented by : q If – Then Statement – If an angle is 40°, then the angle is acute.

E XAMPLE Identify the Hypothesis and Conclusion in the following statements: 1. If a polynomial has six sides, then it is a hexagon. H: A polygon has 6 sides C: it is a hexagon 2. Tamika will advance to the next level of play if she completes the maze in her computer game. H: Tamika Completes the maze in her computer game. C: She will advance to the next level of play. pq

F ORMS OF C ONDITIONAL S TATEMENTS Conditional Statements: Formed By: Given Hypothesis and Conclusion. Symbols: p → q Examples: If two angles have the same measure then they are congruent.

F ORMS OF C ONDITIONAL S TATEMENTS Converse: Formed By: Exchanging Hypothesis and conclusion of the conditional. Symbols: q → p Examples: If two angles are congruent then they have the same measure.

F ORMS OF C ONDITIONAL S TATEMENTS Inverse: Formed By: Negating both the Hypothesis and conclusion of the conditional. Symbols: ~p →~q Examples: If two angles do not have the same measure they are not congruent.

F ORMS OF C ONDITIONAL S TATEMENTS Contra - positive: Formed By: Negating both the Hypothesis and conclusion of the Converse statement. Symbols: ~q →~p Examples: If two angles are not congruent then they do not have the same measure.

Logically Equivalent Statements - are statements with the same truth values. Example: Write the converse, inverse and contra - positive of the following statement: Conditional: If a shape is a square, then it is a rectangle. Converse: If a shape is a rectangle, then it is a square. Inverse: If a shape is not a square, then it is not a rectangle. Contra-positive: If a shape is not a rectangle, then it is not a square.

T RY T HIS : Example: Write the converse, inverse and contra - positive of the following statement: Conditional: If two angles form a linear pair, then they are supplementary. Converse: Inverse: Contra – positive:

Lesson 2-2: Logic 19 V ENN DIAGRAMS : show relationships between different sets of data. can represent conditional statements. is usually drawn as a circle. Every point IN the circle belongs to that set. Every point OUT of the circle does not. Example: A =poodle... a dog B= horse... NOT a dog. B DOGS.A.A...B   dog

Lesson 2-2: Logic 20 F OR ALL..., EVERY..., IF... THEN... All right angles are congruent. Congruent Angles Right Angles Example1: Example 2:Every rose is a flower. Flower Rose Example 3:If two lines are parallel, then they do not intersect. lines that do not intersect parallel lines

Lesson 2-2: Logic 21 T O S HOW R ELATIONSHIPS USING V ENN D IAGRAMS : Blue or Brown (includes Purple) …A  B AB A  B

T HE V ENN D IAGRAM SHOWS THE NUMBER OF STUDENTS ENROLLED IN M ONIQUES ’ DANCE SCHOOL FOR TAP, JAZZ AND BALLET CLASSES a) How Many students are in all three classes? b) How many in tap or ballet? c) How many are in jazz and ballet but not tap? Tap Jazz Ballet