A small town consisting of 2732 households requires pet licenses for dogs and cats. Group 1: the set of households that have purchased dog licenses, Group 2: the set of households that have purchased cat licenses. The town records show that 1821 households have purchased dog licenses while 983 households have purchased cat licenses. Among the households that have purchased dog licenses 1167 have not purchased cat licenses. Determine the cardinality of each of the following sets, including an explanation of the process for how you determined the cardinality:

•   elements in both Group 1 and Group 2

•   elements in Group 2 but not Group 1

•   elements in Group 1 or Group 2

•   elements in neither Group 1 nor Group 2

Let's define the following:

- \( G_1 \): The set of households that have purchased dog licenses (Group 1).
- \( G_2 \): The set of households that have purchased cat licenses (Group 2).
- \( |G_1| = 1821 \): The number of households with dog licenses.
- \( |G_2| = 983 \): The number of households with cat licenses.
- The number of households that have only purchased dog licenses (i.e., those in \( G_1 \) but not in \( G_2 \)) is given as 1167.

We can use these values to determine the cardinality of the required sets using set operations and the principle of inclusion and exclusion.

### 1. **Elements in both Group 1 and Group 2 (households with both dog and cat licenses)**

We are asked to find the number of elements that are in both \( G_1 \) and \( G_2 \) (denoted as \( |G_1 \cap G_2| \)).

We know:
- Households with dog licenses: \( |G_1| = 1821 \)
- Households with only dog licenses (no cat licenses): 1167.

Thus, the number of households that have both dog and cat licenses is:
\[
|G_1 \cap G_2| = |G_1| - \text{Households with only dog licenses} = 1821 - 1167 = 654.
\]

### 2. **Elements in Group 2 but not in Group 1 (households with only cat licenses)**

This is the number of elements in \( G_2 \) but not in \( G_1 \), denoted as \( |G_2 \setminus G_1| \).

We know:
- Total households with cat licenses: \( |G_2| = 983 \).
- Households with both dog and cat licenses: \( |G_1 \cap G_2| = 654 \) (from part 1).

Thus, the number of households with only cat licenses is:
\[
|G_2 \setminus G_1| = |G_2| - |G_1 \cap G_2| = 983 - 654 = 329.
\]

### 3. **Elements in Group 1 or Group 2 (households with either dog or cat licenses or both)**

This is the union of \( G_1 \) and \( G_2 \), denoted as \( |G_1 \cup G_2| \). We can find this using the inclusion-exclusion principle:
\[
|G_1 \cup G_2| = |G_1| + |G_2| - |G_1 \cap G_2|.
\]

Substituting the known values:
\[
|G_1 \cup G_2| = 1821 + 983 - 654 = 2150.
\]

### 4. **Elements in neither Group 1 nor Group 2 (households with neither dog nor cat licenses)**

To find the number of households that have neither dog nor cat licenses, we subtract the number of households that have either (or both) licenses from the total number of households (2732):
\[
\text{Households in neither} = 2732 - |G_1 \cup G_2| = 2732 - 2150 = 582.
\]

### Summary of Results:
- Households with both dog and cat licenses (\( |G_1 \cap G_2| \)): **654**.
- Households with only cat licenses (\( |G_2 \setminus G_1| \)): **329**.
- Households with either dog or cat licenses or both (\( |G_1 \cup G_2| \)): **2150**.
- Households with neither dog nor cat licenses: **582**.

Would you like further explanation on any part of this solution?

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Here are five related questions to deepen understanding:

1. How would the problem change if we had more sets (e.g., households with bird licenses)?
2. What would the result be if the total number of households changed to 3000?
3. How can the inclusion-exclusion principle be extended to three sets?
4. How many households have only dog licenses?
5. How can Venn diagrams help visualize this problem?

**Tip:** Always organize information with set notation and use the inclusion-exclusion principle when dealing with overlapping sets.

A small town consisting of 2732 households requires pet licenses for dogs and cats. Group 1: the set of households that have purchased dog licenses, Group 2: the set of households that have purchased cat licenses. The town records show that 1821 households have purchased dog licenses while 983 households have purchased cat licenses. Among the households that have purchased dog licenses 1167 have not purchased cat licenses. Determine the cardinality of each of the following sets, including an explanation of the process for how you determined the cardinality: • elements in both Group 1 and Group 2 • elements in Group 2 but not Group 1 • elements in Group 1 or Group 2 • elements in neither Group 1 nor Group 2

Learn how to determine the number of households that purchased dog and cat licenses in a town of 2732 households using set theory and the inclusion-exclusion principle. This problem involves calculating households in various categories based on the given data. Suitable for Grades 9-12.

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