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Logic 1. To write a conditional 2. To identify the hypothesis and conclusion in a conditional 3. To write the converse, inverse and contrapositive of a given conditional 4. To state the truth value of each of the above (draw conclusions) 5. To write a biconditional Conditional- an if-then statement Write a conditional with each of the following: • • • • • • A right angle has a measure = 90◦. If an angle is a rt. <, then it = 90◦. If an < = 90◦, then it is a rt. <. Christmas is on December 25th. If it is Christmas, then it is Dec. 25th. If it is Dec. 25th, then it is Christmas. Every conditional has a hypothesis and a conclusion. The hypothesis always follows the if and the conclusion always follows the then. Underline the hypothesis once and the conclusion twice for the previous statements. Conditional- an if-then statement Write a conditional with each of the following: • • • • • • A right angle has a measure = 90◦. If an angle is a rt. <, then it = 90◦. If an < = 90◦, then it is a rt. <. Christmas is on December 25th. If it is Christmas, then it is Dec. 25th. If it is Dec. 25th, then it is Christmas. The following is a Venn diagram. Use it to write a conditional. At least a 4 year college degree Teacher If you are a teacher, then you have at least a 4 year college degree. Write a conditional. dogs Chow If you are a chow, then you are a dog. Counterexamples-examples for which a conjecture (statement) is incorrect. If it is a weekday, then it is Monday. counterexample– it could be Tuesday If the animal is a dog, then it is a poodle. counterexample--- it could be a lab If a number is prime it is not even. counterexample---2 is a prime # Define converse, inverse, and contrapositive of a given conditional. Converse of a conditional ----flips the hypothesis and conclusion Inverse of a conditional-----negates both the hypothesis and conclusion Contrapositive of a conditional ----flips and negates the conditional Logic Symbols • Conditional p→q • Converse q→p Flips conditional • Inverse ~p → ~q negates conditional • Contrapositive ~q → ~p flips and negates conditional If 2 segments are congruent, then they are equal in length. • Write the converse, inverse,& contrapositive for the above statement. Converse---- If 2 segments are equal in length, then they are congruent. Inverse-----If 2 segments are not congruent, then they are not equal in length. Contrapositive---- If 2 segments are not equal in length, then they are not congruent. If 2 angles are vertical, then they are congruent. • Write the 1.converse 2. inverse 3. contrapositive. • If 2 angles are congruent, then they are vertical. • If 2 angles are not vertical, then they are not congruent. • If 2 angles are not congruent, then they are not vertical. Write the 1.converse 2. inverse 3. contrapositive of the following definition If an angle is a right angle, then the angle is equal to 90 degrees. If an angle is equal to 90 degrees, then it is a right angle. If an angle is not a right angle, then it is not equal to 90 degrees. If an angle is not equal to 90 degrees then it is not a right angle. Go back and determine the truth values of all your problems. Do you notice anything? If 2 segments are congruent, then they are equal in length. • Write the converse, inverse,& contrapositive for the above statement. Converse---- If 2 segments are equal in length, then they are congruent. Inverse-----If 2 segments are not congruent, then they are not equal in length. Contrapositive---- If 2 segments are not equal in length, then they are not congruent. Note the above is a definition!!!! If 2 angles are vertical, then they are congruent. • Write the 1.converse 2. inverse 3. contrapositive. • If 2 angles are congruent, then they are vertical. • If 2 angles are not vertical, then they are not congruent. • If 2 angles are not congruent, then they are not vertical. • Note the above is a theorem!!!! Write the 1.converse 2. inverse 3. contrapositive of the following definition If an angle is a right angle, then the angle is equal to 90 degrees. If an angle is equal to 90 degrees, then it is a right angle. If an angle is not a right angle, then it is not equal to 90 degrees. If an angle is not equal to 90 degrees then it is not a right angle. Truth Values • The conditional and the contrapositive always have the same truth value. • The converse and the inverse always have the same truth value. Truth Values • Note the truth values are all true if your conditional started with a definition. • This is not necessarily true for a theorem. D B Isosceles Triangle Theorem A If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. If DB ≅ DA then, <B ≅ < A. Converse of D B Isosceles Triangle Theorem A If 2 <‘s of a triangle are congruent, then the sides opposite those angles are congruent. If <B ≅ < A, then BD ≅ DA. Biconditional- a statement that combines a true conditional with its true converse in an if and only if statement. Conditional- If an < is a rt <, then it = 90◦ converse If an < = 90◦, then it is a right <. • An angle is a right angle if and only if it is equal to 90 degrees. • An angle is equal to 90 degrees iff it is a right angle. Write a biconditional. • If 3 points lie on the same line, then they are collinear. • If 3 points are collinear, then they lie on the same line. • 3 points are collinear if and only if they lie on the same line • 3 points are on the same line if and only if they are collinear. Write a biconditional. • If 2 lines are skew, then they are noncoplanar. • If 2 lines are noncoplanar, then they are skew. • 2 lines are noncoplanar iff they are skew. • 2 lines are skew iff they are noncoplanar. Write a converse, inverse, contrapositive and biconditional for the following: If 2n = 8, then 3n = 12. Converse If 3n = 12, then 2n = 8. Inverse If 2n ≠ 8, then 3n ≠ 12. Contrapositive If 3n ≠ 12, then 2n ≠ 8. 2n = 8 iff 3n = 12 3n = 12 iff 2n = 8 •Note every definition is biconditional! Rewrite as 2 if-then statements (conditional and converse) (x+4) ( x-5) = 0 iff x= -4 or x= 5 If (x+4) (x-5) = 0 then x= -4 or x= 5. If x = -4 or x = 5, then (x+4) ( x-5) = 0. Write the converse of the given conditional, then write 2 biconditionals • 1. If a point is a midpoint, then it divides a segment into 2 congruent halves. If a point divides a segment into 2 ¤ halves, then it is a midpoint. A pt. is a midpt iff it divides a segment into 2 ¤ halves. A pt. divides a segment into 2 ¤ halves iff it is a midpoint. Assignments Homework---pp.71-73 (2-4;9-12;15-29;3335) p. 78 (1-11 0dd) p 267 (1-9 odd) Classwork– HM worksheet # 11