Practice exam worksheet focusing on logical statements and their converse, inverse, and contrapositive forms.
Practice exam worksheet titled "Practice Exam in Converse, Inverse and Contrapositive of an Implication" with ten statements to convert into their converse, inverse, and contrapositive forms.
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Step-by-step solution for: SOLUTION: Ppractice exam in converse inverse and contrapositive of ...
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Show Answer Key & Explanations
Step-by-step solution for: SOLUTION: Ppractice exam in converse inverse and contrapositive of ...
You're working on a practice exam about converse, inverse, and contrapositive of conditional (implication) statements.
Let’s first review the definitions:
Given a conditional statement:
> If P, then Q → written as P → Q
- Converse: Switch P and Q → If Q, then P → Q → P
- Inverse: Negate both P and Q → If not P, then not Q → ¬P → ¬Q
- Contrapositive: Switch and negate both → If not Q, then not P → ¬Q → ¬P
Important note: The contrapositive is logically equivalent to the original statement. The converse and inverse are logically equivalent to each other, but not necessarily to the original.
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Now let’s solve each problem step by step.
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- P: You are in love
- Q: You are inspired
✔ Converse: If you are inspired, then you are in love.
✔ Inverse: If you are not in love, then you are not inspired.
✔ Contrapositive: If you are not inspired, then you are not in love.
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- P: Two lines are parallel
- Q: Two lines do not intersect
✔ Converse: If two lines do not intersect, then they are parallel.
*(Note: In Euclidean geometry, this is true — non-intersecting lines are parallel.)*
✔ Inverse: If two lines are not parallel, then they intersect.
✔ Contrapositive: If two lines intersect, then they are not parallel.
---
- P: Number is divisible by 2 and 3
- Q: Number is divisible by 6
✔ Converse: If the number is divisible by 6, then it is divisible by 2 and 3.
*(True — since 6 = 2×3, divisibility by 6 implies divisibility by both 2 and 3.)*
✔ Inverse: If the number is not divisible by 2 and 3, then it is not divisible by 6.
*(Note: “Not divisible by 2 and 3” means NOT (div by 2 AND div by 3) → which is equivalent to: not div by 2 OR not div by 3. So if either fails, then not div by 6 — which is true.)*
✔ Contrapositive: If the number is not divisible by 6, then it is not divisible by 2 and 3.
*(Same logic as inverse — true.)*
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*(Note: Should probably say “garbage is”, but we’ll keep as given.)*
- P: Garbage are disposed properly
- Q: Diseases will be prevented
✔ Converse: If diseases will be prevented, then garbage are disposed properly.
✔ Inverse: If garbage are not disposed properly, then diseases will not be prevented.
✔ Contrapositive: If diseases will not be prevented, then garbage are not disposed properly.
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- P: You participate in class
- Q: You will get extra points
✔ Converse: If you get extra points, then you participated in class.
✔ Inverse: If you do not participate in class, then you will not get extra points.
✔ Contrapositive: If you do not get extra points, then you did not participate in class.
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- P: Moon is full
- Q: Vampires are prowling
✔ Converse: If the vampires are prowling, then the moon is full.
✔ Inverse: If the moon is not full, then the vampires are not prowling.
✔ Contrapositive: If the vampires are not prowling, then the moon is not full.
*(Note: This is fictional — logic still applies regardless of truth value.)*
---
- P: You do not understand geometry
- Q: You do not know how to reason deductively
✔ Converse: If you do not know how to reason deductively, then you do not understand geometry.
✔ Inverse: If you understand geometry, then you know how to reason deductively.
*(Negation of P: “you understand geometry”; negation of Q: “you know how to reason deductively”)*
✔ Contrapositive: If you know how to reason deductively, then you understand geometry.
---
- P: You do not have a sense of balance
- Q: You cannot skateboard
✔ Converse: If you cannot skateboard, then you do not have a sense of balance.
✔ Inverse: If you have a sense of balance, then you can skateboard.
✔ Contrapositive: If you can skateboard, then you have a sense of balance.
---
- P: It rains
- Q: I do not go fishing
✔ Converse: If I do not go fishing, then it rains.
✔ Inverse: If it does not rain, then I go fishing.
✔ Contrapositive: If I go fishing, then it does not rain.
---
- P: You read the question carefully
- Q: The solution is easy
✔ Converse: If the solution is easy, then you read the question carefully.
✔ Inverse: If you do not read the question carefully, then the solution is not easy.
✔ Contrapositive: If the solution is not easy, then you did not read the question carefully.
---
## ✔ Final Answers Summary:
---
1.
- Converse: If you are inspired, then you are in love.
- Inverse: If you are not in love, then you are not inspired.
- Contrapositive: If you are not inspired, then you are not in love.
2.
- Converse: If two lines do not intersect, then they are parallel.
- Inverse: If two lines are not parallel, then they intersect.
- Contrapositive: If two lines intersect, then they are not parallel.
3.
- Converse: If the number is divisible by 6, then it is divisible by 2 and 3.
- Inverse: If the number is not divisible by 2 and 3, then it is not divisible by 6.
- Contrapositive: If the number is not divisible by 6, then it is not divisible by 2 and 3.
4.
- Converse: If diseases will be prevented, then garbage are disposed properly.
- Inverse: If garbage are not disposed properly, then diseases will not be prevented.
- Contrapositive: If diseases will not be prevented, then garbage are not disposed properly.
5.
- Converse: If you get extra points, then you participated in class.
- Inverse: If you do not participate in class, then you will not get extra points.
- Contrapositive: If you do not get extra points, then you did not participate in class.
6.
- Converse: If the vampires are prowling, then the moon is full.
- Inverse: If the moon is not full, then the vampires are not prowling.
- Contrapositive: If the vampires are not prowling, then the moon is not full.
7.
- Converse: If you do not know how to reason deductively, then you do not understand geometry.
- Inverse: If you understand geometry, then you know how to reason deductively.
- Contrapositive: If you know how to reason deductively, then you understand geometry.
8.
- Converse: If you cannot skateboard, then you do not have a sense of balance.
- Inverse: If you have a sense of balance, then you can skateboard.
- Contrapositive: If you can skateboard, then you have a sense of balance.
9.
- Converse: If I do not go fishing, then it rains.
- Inverse: If it does not rain, then I go fishing.
- Contrapositive: If I go fishing, then it does not rain.
10.
- Converse: If the solution is easy, then you read the question carefully.
- Inverse: If you do not read the question carefully, then the solution is not easy.
- Contrapositive: If the solution is not easy, then you did not read the question carefully.
---
Let me know if you’d like these formatted for printing or turned into a worksheet! 📝
Let’s first review the definitions:
Given a conditional statement:
> If P, then Q → written as P → Q
- Converse: Switch P and Q → If Q, then P → Q → P
- Inverse: Negate both P and Q → If not P, then not Q → ¬P → ¬Q
- Contrapositive: Switch and negate both → If not Q, then not P → ¬Q → ¬P
Important note: The contrapositive is logically equivalent to the original statement. The converse and inverse are logically equivalent to each other, but not necessarily to the original.
---
Now let’s solve each problem step by step.
---
1. If you are in love, then you are inspired.
- P: You are in love
- Q: You are inspired
✔ Converse: If you are inspired, then you are in love.
✔ Inverse: If you are not in love, then you are not inspired.
✔ Contrapositive: If you are not inspired, then you are not in love.
---
2. If two lines are parallel, then the two lines do not intersect.
- P: Two lines are parallel
- Q: Two lines do not intersect
✔ Converse: If two lines do not intersect, then they are parallel.
*(Note: In Euclidean geometry, this is true — non-intersecting lines are parallel.)*
✔ Inverse: If two lines are not parallel, then they intersect.
✔ Contrapositive: If two lines intersect, then they are not parallel.
---
3. If the number is divisible by 2 and 3, then the number is divisible by 6.
- P: Number is divisible by 2 and 3
- Q: Number is divisible by 6
✔ Converse: If the number is divisible by 6, then it is divisible by 2 and 3.
*(True — since 6 = 2×3, divisibility by 6 implies divisibility by both 2 and 3.)*
✔ Inverse: If the number is not divisible by 2 and 3, then it is not divisible by 6.
*(Note: “Not divisible by 2 and 3” means NOT (div by 2 AND div by 3) → which is equivalent to: not div by 2 OR not div by 3. So if either fails, then not div by 6 — which is true.)*
✔ Contrapositive: If the number is not divisible by 6, then it is not divisible by 2 and 3.
*(Same logic as inverse — true.)*
---
4. If garbage are disposed properly, then the diseases will be prevented.
*(Note: Should probably say “garbage is”, but we’ll keep as given.)*
- P: Garbage are disposed properly
- Q: Diseases will be prevented
✔ Converse: If diseases will be prevented, then garbage are disposed properly.
✔ Inverse: If garbage are not disposed properly, then diseases will not be prevented.
✔ Contrapositive: If diseases will not be prevented, then garbage are not disposed properly.
---
5. If you participate in class, then you will get extra points.
- P: You participate in class
- Q: You will get extra points
✔ Converse: If you get extra points, then you participated in class.
✔ Inverse: If you do not participate in class, then you will not get extra points.
✔ Contrapositive: If you do not get extra points, then you did not participate in class.
---
6. If the moon is full, then the vampires are prowling.
- P: Moon is full
- Q: Vampires are prowling
✔ Converse: If the vampires are prowling, then the moon is full.
✔ Inverse: If the moon is not full, then the vampires are not prowling.
✔ Contrapositive: If the vampires are not prowling, then the moon is not full.
*(Note: This is fictional — logic still applies regardless of truth value.)*
---
7. If you do not understand geometry, then you do not know how to reason deductively.
- P: You do not understand geometry
- Q: You do not know how to reason deductively
✔ Converse: If you do not know how to reason deductively, then you do not understand geometry.
✔ Inverse: If you understand geometry, then you know how to reason deductively.
*(Negation of P: “you understand geometry”; negation of Q: “you know how to reason deductively”)*
✔ Contrapositive: If you know how to reason deductively, then you understand geometry.
---
8. If you do not have a sense of balance, then you cannot skateboard.
- P: You do not have a sense of balance
- Q: You cannot skateboard
✔ Converse: If you cannot skateboard, then you do not have a sense of balance.
✔ Inverse: If you have a sense of balance, then you can skateboard.
✔ Contrapositive: If you can skateboard, then you have a sense of balance.
---
9. If it rains, then I do not go fishing.
- P: It rains
- Q: I do not go fishing
✔ Converse: If I do not go fishing, then it rains.
✔ Inverse: If it does not rain, then I go fishing.
✔ Contrapositive: If I go fishing, then it does not rain.
---
10. If you read the question carefully, then the solution is easy.
- P: You read the question carefully
- Q: The solution is easy
✔ Converse: If the solution is easy, then you read the question carefully.
✔ Inverse: If you do not read the question carefully, then the solution is not easy.
✔ Contrapositive: If the solution is not easy, then you did not read the question carefully.
---
## ✔ Final Answers Summary:
---
1.
- Converse: If you are inspired, then you are in love.
- Inverse: If you are not in love, then you are not inspired.
- Contrapositive: If you are not inspired, then you are not in love.
2.
- Converse: If two lines do not intersect, then they are parallel.
- Inverse: If two lines are not parallel, then they intersect.
- Contrapositive: If two lines intersect, then they are not parallel.
3.
- Converse: If the number is divisible by 6, then it is divisible by 2 and 3.
- Inverse: If the number is not divisible by 2 and 3, then it is not divisible by 6.
- Contrapositive: If the number is not divisible by 6, then it is not divisible by 2 and 3.
4.
- Converse: If diseases will be prevented, then garbage are disposed properly.
- Inverse: If garbage are not disposed properly, then diseases will not be prevented.
- Contrapositive: If diseases will not be prevented, then garbage are not disposed properly.
5.
- Converse: If you get extra points, then you participated in class.
- Inverse: If you do not participate in class, then you will not get extra points.
- Contrapositive: If you do not get extra points, then you did not participate in class.
6.
- Converse: If the vampires are prowling, then the moon is full.
- Inverse: If the moon is not full, then the vampires are not prowling.
- Contrapositive: If the vampires are not prowling, then the moon is not full.
7.
- Converse: If you do not know how to reason deductively, then you do not understand geometry.
- Inverse: If you understand geometry, then you know how to reason deductively.
- Contrapositive: If you know how to reason deductively, then you understand geometry.
8.
- Converse: If you cannot skateboard, then you do not have a sense of balance.
- Inverse: If you have a sense of balance, then you can skateboard.
- Contrapositive: If you can skateboard, then you have a sense of balance.
9.
- Converse: If I do not go fishing, then it rains.
- Inverse: If it does not rain, then I go fishing.
- Contrapositive: If I go fishing, then it does not rain.
10.
- Converse: If the solution is easy, then you read the question carefully.
- Inverse: If you do not read the question carefully, then the solution is not easy.
- Contrapositive: If the solution is not easy, then you did not read the question carefully.
---
Let me know if you’d like these formatted for printing or turned into a worksheet! 📝
Parent Tip: Review the logic above to help your child master the concept of inverse converse contrapositive worksheet.
