Formal Logic: Conditional Statements worksheet with 20 examples for students to analyze and determine truth values.
A worksheet titled "Formal Logic: Conditional Statements" with instructions and 20 numbered statements for students to identify conditional statements and their truth values. The page includes a cartoon drawing of a person thinking with a light bulb above their head. Created by Creative Math Lessons, 2015.
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Show Answer Key & Explanations
Step-by-step solution for: Geometry: Conditional Statements III - Converse, Inverse, Contrapositives
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Show Answer Key & Explanations
Step-by-step solution for: Geometry: Conditional Statements III - Converse, Inverse, Contrapositives
To solve the problem, we need to determine the truth value of each conditional statement provided. A conditional statement is written in the form "If \( P \), then \( Q \)" (symbolically: \( P \rightarrow Q \)). The truth value of a conditional statement depends on the truth values of its components:
- True if:
- \( P \) is true and \( Q \) is true.
- \( P \) is false (regardless of \( Q \)).
- False only if:
- \( P \) is true and \( Q \) is false.
For each statement, we will analyze whether it is always true or provide a counterexample if it is not.
---
- Analysis: This statement is generally true because horses are known to have four legs. However, there could be exceptions (e.g., a deformed horse with fewer legs). Since the statement does not account for such exceptions, it is not universally true.
- Counterexample: A horse born with three legs due to a birth defect.
- Truth Value: False.
---
- Analysis: Oranges are indeed a type of fruit. There are no exceptions to this rule.
- Truth Value: True.
---
- Analysis: This statement is nonsensical because "setting a chair" is not related to eating a vegetable. It is logically false because the premise ("lunch is setting a chair") is absurd.
- Truth Value: False.
---
- Analysis: This statement is vague because "might win a million dollars" is not guaranteed by being next to the speaker. The truth of the conclusion depends on external factors not mentioned in the premise.
- Counterexample: You are standing next to the speaker, but no lottery or contest is happening.
- Truth Value: False.
---
- Analysis: Vegetables contain calories, so eating them means consuming calories. This statement is true under normal circumstances.
- Truth Value: True.
---
- Analysis: Soda contains calories, so drinking a lot of soda means consuming calories. This statement is true.
- Truth Value: True.
---
- Analysis: The color of octagons has no logical connection to the reality of the moon. The truth of the conclusion ("the moon is real") is independent of the premise.
- Counterexample: Octagons can be any color, but the moon's existence is unrelated.
- Truth Value: False.
---
- Analysis: The existence of Pythagoras in ancient Greece is a historical fact. If you exist, it does not affect the historical truth about Pythagoras.
- Truth Value: True.
---
- Analysis: The Empire State Building is located in New York City. Therefore, if Walter is at the Empire State Building, he must be in New York City.
- Truth Value: True.
---
- Analysis: This statement is grammatically incorrect and unclear. It seems to compare the cloudiness of Athens to Greece, which is illogical. The statement is false due to its lack of clarity.
- Truth Value: False.
---
- Analysis: The Tonight Show airs on Fridays, so if it is Friday, it is reasonable to conclude that it is The Tonight Show (assuming the context is correct).
- Truth Value: True.
---
- Analysis: Thursdays are indeed weekdays. This statement is true.
- Truth Value: True.
---
- Analysis: By definition, two angles are complementary if their measures add up to 90 degrees. This statement is true.
- Truth Value: True.
---
- Analysis: Vertical angles are always congruent by definition. This statement is true.
- Truth Value: True.
---
- Analysis: Eating a frozen French Bulldog (which is absurd) does not imply eating chocolate. The statement is false because the premise is nonsensical.
- Truth Value: False.
---
- Analysis: Eating a frozen French Bulldog is absurd and unrelated to being in California. The statement is false.
- Truth Value: False.
---
- Analysis: This is a repetition of the previous statement. It is false for the same reason.
- Truth Value: False.
---
- Analysis: Eating blueberries does not imply eating a live mouse. The statement is false.
- Truth Value: False.
---
- Analysis: South America includes many countries besides Brazil (e.g., Argentina, Colombia). The statement is false because being in South America does not necessarily mean being in Brazil.
- Counterexample: He is in Argentina.
- Truth Value: False.
---
- Analysis: The number "3" is arbitrary and unrelated to whether it is an election year. The statement is false.
- Counterexample: The year 2023 is not an election year.
- Truth Value: False.
---
\[
\boxed{
\begin{array}{ll}
1. & \text{False} \\
2. & \text{True} \\
3. & \text{False} \\
4. & \text{False} \\
5. & \text{True} \\
6. & \text{True} \\
7. & \text{False} \\
8. & \text{True} \\
9. & \text{True} \\
10. & \text{False} \\
11. & \text{True} \\
12. & \text{True} \\
13. & \text{True} \\
14. & \text{True} \\
15. & \text{False} \\
16. & \text{False} \\
17. & \text{False} \\
18. & \text{False} \\
19. & \text{False} \\
20. & \text{False} \\
\end{array}
}
\]
- True if:
- \( P \) is true and \( Q \) is true.
- \( P \) is false (regardless of \( Q \)).
- False only if:
- \( P \) is true and \( Q \) is false.
For each statement, we will analyze whether it is always true or provide a counterexample if it is not.
---
1. If an animal is a horse, then it has four legs.
- Analysis: This statement is generally true because horses are known to have four legs. However, there could be exceptions (e.g., a deformed horse with fewer legs). Since the statement does not account for such exceptions, it is not universally true.
- Counterexample: A horse born with three legs due to a birth defect.
- Truth Value: False.
---
2. If food is an orange, then it is fruit.
- Analysis: Oranges are indeed a type of fruit. There are no exceptions to this rule.
- Truth Value: True.
---
3. If lunch is setting a chair, then it is eating a vegetable.
- Analysis: This statement is nonsensical because "setting a chair" is not related to eating a vegetable. It is logically false because the premise ("lunch is setting a chair") is absurd.
- Truth Value: False.
---
4. If you are standing next to me, then you might win a million dollars.
- Analysis: This statement is vague because "might win a million dollars" is not guaranteed by being next to the speaker. The truth of the conclusion depends on external factors not mentioned in the premise.
- Counterexample: You are standing next to the speaker, but no lottery or contest is happening.
- Truth Value: False.
---
5. If you eat vegetables, then you are consuming calories.
- Analysis: Vegetables contain calories, so eating them means consuming calories. This statement is true under normal circumstances.
- Truth Value: True.
---
6. If someone is drinking a lot of soda, then he is consuming calories.
- Analysis: Soda contains calories, so drinking a lot of soda means consuming calories. This statement is true.
- Truth Value: True.
---
7. If octagons are red, then the moon is real.
- Analysis: The color of octagons has no logical connection to the reality of the moon. The truth of the conclusion ("the moon is real") is independent of the premise.
- Counterexample: Octagons can be any color, but the moon's existence is unrelated.
- Truth Value: False.
---
8. If you exist, then Pythagoras existed in ancient Greece.
- Analysis: The existence of Pythagoras in ancient Greece is a historical fact. If you exist, it does not affect the historical truth about Pythagoras.
- Truth Value: True.
---
9. If Walter is at the Empire State Building, then Walter is in New York City.
- Analysis: The Empire State Building is located in New York City. Therefore, if Walter is at the Empire State Building, he must be in New York City.
- Truth Value: True.
---
10. If it is cloudy, then it is city Athens than in Greece.
- Analysis: This statement is grammatically incorrect and unclear. It seems to compare the cloudiness of Athens to Greece, which is illogical. The statement is false due to its lack of clarity.
- Truth Value: False.
---
11. If it is Friday, then it is The Tonight Show.
- Analysis: The Tonight Show airs on Fridays, so if it is Friday, it is reasonable to conclude that it is The Tonight Show (assuming the context is correct).
- Truth Value: True.
---
12. If it is Thursday, then it is a weekday.
- Analysis: Thursdays are indeed weekdays. This statement is true.
- Truth Value: True.
---
13. If two angles are complementary, then their measures add up to 90 degrees.
- Analysis: By definition, two angles are complementary if their measures add up to 90 degrees. This statement is true.
- Truth Value: True.
---
14. If two angles are vertical angles, then they are congruent.
- Analysis: Vertical angles are always congruent by definition. This statement is true.
- Truth Value: True.
---
15. If he is eating a frozen French Bulldog, then he is eating chocolate.
- Analysis: Eating a frozen French Bulldog (which is absurd) does not imply eating chocolate. The statement is false because the premise is nonsensical.
- Truth Value: False.
---
16. If he is eating a frozen French Bulldog, then he is in California.
- Analysis: Eating a frozen French Bulldog is absurd and unrelated to being in California. The statement is false.
- Truth Value: False.
---
17. If he is eating a frozen French Bulldog, then he is in California.
- Analysis: This is a repetition of the previous statement. It is false for the same reason.
- Truth Value: False.
---
18. If he is eating blueberries, then he is eating a live mouse.
- Analysis: Eating blueberries does not imply eating a live mouse. The statement is false.
- Truth Value: False.
---
19. If he is in South America, then he is in Brazil.
- Analysis: South America includes many countries besides Brazil (e.g., Argentina, Colombia). The statement is false because being in South America does not necessarily mean being in Brazil.
- Counterexample: He is in Argentina.
- Truth Value: False.
---
20. If it is 3, then it is an election year.
- Analysis: The number "3" is arbitrary and unrelated to whether it is an election year. The statement is false.
- Counterexample: The year 2023 is not an election year.
- Truth Value: False.
---
Final Answer
\[
\boxed{
\begin{array}{ll}
1. & \text{False} \\
2. & \text{True} \\
3. & \text{False} \\
4. & \text{False} \\
5. & \text{True} \\
6. & \text{True} \\
7. & \text{False} \\
8. & \text{True} \\
9. & \text{True} \\
10. & \text{False} \\
11. & \text{True} \\
12. & \text{True} \\
13. & \text{True} \\
14. & \text{True} \\
15. & \text{False} \\
16. & \text{False} \\
17. & \text{False} \\
18. & \text{False} \\
19. & \text{False} \\
20. & \text{False} \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of logic statements worksheet.
