Question

write each biconditional as a conjunction of two conditionals. 16. you will make the team if and only if you practice diligently. 17. you will go to heaven if and only if you are saved. 18. the roads will be slick if and only if there is ice on them. 19. a figure is a regular polygon if and only if it is both equilateral

Explanation:

Step1: Recall biconditional - conditional relationship

A biconditional "p if and only if q" is equivalent to the conjunction of "if p then q" and "if q then p".

Step2: Translate 16

Let p = "You will make the team" and q = "You practice diligently". The biconditional can be written as "If you will make the team, then you practice diligently and if you practice diligently, then you will make the team".

Step3: Translate 17

Let p = "You will go to heaven" and q = "You are saved". The biconditional can be written as "If you will go to heaven, then you are saved and if you are saved, then you will go to heaven".

Step4: Translate 18

Let p = "The roads will be slick" and q = "There is ice on them". The biconditional can be written as "If the roads will be slick, then there is ice on them and if there is ice on them, then the roads will be slick".

Step5: Translate 19 (incomplete in the given text but conceptually)

Let p = "A figure is a regular polygon" and assume the full q = "it is both equilateral and equiangular". It can be written as "If a figure is a regular polygon, then it is both equilateral and equiangular and if it is both equilateral and equiangular, then a figure is a regular polygon".

Answer:

1. If you will make the team, then you practice diligently and if you practice diligently, then you will make the team.
2. If you will go to heaven, then you are saved and if you are saved, then you will go to heaven.
3. If the roads will be slick, then there is ice on them and if there is ice on them, then the roads will be slick.
4. If a figure is a regular polygon, then it is both equilateral and equiangular and if it is both equilateral and equiangular, then a figure is a regular polygon.