AMSCO'S Geometry. New York - Rye High School

80 Logic17. If 2b 6 14, then 2b 8. 18. If it is 8:15 A.M., then it is morning.If 2b 8, then b 4.It is not morning.2b 6 1419. If k is a prime, then k 8. 20. x is even and a prime if and only if x 2.k = 8 x 221. It is February or March, and it is not 22. If x is divisible by 4, then x is divisiblesummer. by 2.It is not March. x is divisible by 2.23. On Saturdays, we go bowling or we 24. I study computer science, and wood shopfly kites.or welding.Last Saturday, we did not go bowling.I do not take woodshop.25. Five is a prime if and only if five has 26. If x is an integer greater than 2 and xexactly two factors.is a prime, then x is odd.Five is a prime.x is not an odd integer.27. If a ray bisects an angle, the ray divides the angle into two congruent angles.Ray DF does not divide angle CDE into two congruent angles.2-8 DRAWING CONCLUSIONSEXAMPLE 1Many important decisions as well as everyday choices are made by applyingthe principles of logic. Games and riddles also often depend on logic for theirsolution.The three statements given below are each true. What conclusion can be foundto 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.SolutionA conditional and its contrapositive are logically equivalent. Therefore, “IfRachel does not like to sing, then Rachel will not join the choir” is true.By the Law of Detachment, when the hypothesis of a true conditional is true,the conclusion must be true. Therefore, since “Rachel does not like to sing” istrue, “Rachel will not join the choir” must be true.Since “Rachel will not join the choir” is true, then its negation, “Rachel willjoin the choir,” must be false.