Question: 6. Use the Law of Syllogism to draw a conclusion from the two given statements.
If the shape is a rectangle, then then it is also a parallelogram.
If the shape has four right angles, then it is rectangle.
a. If the shape is a parallelogram, then the shape is a rectangle.
b. If the shape has four right angles, then it is a parallelogram.
c. If the shape is a parallelogram, then it has four right angles.
d. If the shape is a rectangle, then it has four right angles.
7. Use the Law of Detachment to draw a conclusion from the two given statements. If not possible, select not possible.
If you plan to attend Homecoming, then you must buy a ticket.
Justin plans to attend Homecoming.
a. Justin must buy a ticket.
b. Justin doesn’t plan to attend Homecoming.
c. If Justing doesn’t buy a ticket, then he doesn’t plan to attend Homecoming.
d. Not Possible
8. The Venn Diagram shows the elements of the sets U, A, B, and C. (6 points)
A \cup B = {10, 12, 2, 4, 7, 9, 3, 8, 1, 6, 13}
B \cap C = {2, 4, 7}
\sim A \cup C = {3, 5, 8}
9. The Venn Diagram shows facts about the Seniors at Albemarle High School. (6 points)
How many seniors have a driver’s license and take college classes? 24 + 7 = 31
How many seniors take college classes or have a part-time job?
60 + 7 + 18 + 45 + 32 + 24 = 186
How many seniors take college classes but don’t have their driver’s license? 18 + 32 = 50
1. **Problem 6: Law of Syllogism**
Given:
- If the shape is a rectangle, then it is also a parallelogram.
- If the shape has four right angles, then it is a rectangle.
The Law of Syllogism states: If $p \to q$ and $q \to r$, then $p \to r$.
Here:
- $p$: The shape has four right angles.
- $q$: The shape is a rectangle.
- $r$: The shape is a parallelogram.
By the Law of Syllogism:
$$p \to q, \quad q \to r \implies p \to r$$
So:
- If the shape has four right angles, then it is a parallelogram.
**Answer:** b. If the shape has four right angles, then it is a parallelogram.
2. **Problem 7: Law of Detachment**
Given:
- If you plan to attend Homecoming, then you must buy a ticket.
- Justin plans to attend Homecoming.
The Law of Detachment states: If $p \to q$ and $p$ is true, then $q$ is true.
Here:
- $p$: Justin plans to attend Homecoming.
- $q$: Justin must buy a ticket.
Since $p$ is true, we conclude $q$ is true.
**Answer:** a. Justin must buy a ticket.
3. **Problem 8: Venn Diagram Sets**
Given:
- $A \cup B = \{10, 12, 2, 4, 7, 9, 3, 8, 1, 6, 13\}$
- $B \cap C = \{2, 4, 7\}$
- $\sim A \cup C = \{3, 5, 8\}$
This problem provides set relations and elements but does not ask a specific question, so no further calculation is done here.
4. **Problem 9: Seniors at Albemarle High School**
Given:
- Seniors with driver’s license and college classes: $24 + 7 = 31$
- Seniors who take college classes or have a part-time job: $60 + 7 + 18 + 45 + 32 + 24 = 186$
- Seniors who take college classes but don’t have their driver’s license: $18 + 32 = 50$
These are direct sums of given numbers representing counts in the Venn diagram.
**Summary:**
- Problem 6 answer: b
- Problem 7 answer: a
- Problems 8 and 9 are data interpretation with no explicit question to solve here.