Logic To write a conditional

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Presentation transcript:

Logic To write a conditional To identify the hypothesis and conclusion in a conditional To write the converse, inverse and contrapositive of a given conditional To state the truth value of each of the above (draw conclusions) 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 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. If you are a teacher, then you have at least a 4 year college degree. At least a 4 year college degree Teacher

Write a conditional. 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 Inverse ~p → ~q 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 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 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 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.

Isosceles Triangle Theorem D Isosceles Triangle Theorem If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. B A If DB ≅ DA then, <B ≅ < A.

Isosceles Triangle Theorem Converse of Isosceles Triangle Theorem If 2 <‘s of a triangle are congruent, then the sides opposite those angles are congruent. D B A 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;33-35) p. 78 (1-11 0dd) p 267 (1-9 odd) Classwork– HM worksheet # 11