If X and Y are sets of integers, X@Y denotes the set of integers that belong to set X or...

1. Translate the problem requirements: The symbol \(\mathrm{X@Y}\) represents the symmetric difference - integers that are in X or Y, but not in both sets. We need to find how many integers are in this combined set.
2. Identify the components: Break down what we know about the sets - total elements in each set and the overlap between them.
3. Calculate elements unique to each set: Find how many integers belong only to X and only to Y by subtracting the shared elements from each set's total.
4. Combine the unique elements: Add together the elements that are only in X and only in Y to get the final count for \(\mathrm{X@Y}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what \(\mathrm{X@Y}\) means in everyday terms. The symbol \(\mathrm{X@Y}\) represents elements that are "exclusively" in one set or the other - think of it like people who belong to either Club X or Club Y, but not both clubs at the same time.

Imagine you have a group of friends where some only belong to the Chess Club (X), some only belong to the Drama Club (Y), and some belong to both clubs. The \(\mathrm{X@Y}\) operation gives us only those friends who are "loyal" to just one club - either only Chess or only Drama, but not both.

Process Skill: TRANSLATE - Converting the abstract set operation into concrete, relatable terms

2. Identify the components

Now let's organize what we know about our sets:

- Set X has 10 integers total
- Set Y has 18 integers total
- 6 integers appear in both X and Y (the "overlap")

Think of this like a survey: 10 people like pizza, 18 people like burgers, and 6 people like both pizza and burgers. We want to find how many people like exactly one of these foods (pizza only OR burgers only, but not both).

3. Calculate elements unique to each set

To find the elements that belong to only one set, we need to subtract the shared elements from each set's total:

Elements only in X = Total in X - Elements in both
Elements only in X = \(\mathrm{10 - 6 = 4}\)

Elements only in Y = Total in Y - Elements in both
Elements only in Y = \(\mathrm{18 - 6 = 12}\)

Using our food example: People who like only pizza = \(\mathrm{10 - 6 = 4}\) people, and people who like only burgers = \(\mathrm{18 - 6 = 12}\) people.

4. Combine the unique elements

The symmetric difference \(\mathrm{X@Y}\) consists of all elements that are in exactly one of the sets. This means we add together:

- Elements only in X: 4
- Elements only in Y: 12

Therefore: \(\mathrm{X@Y = 4 + 12 = 16}\)

In our food example, the total number of people who like exactly one food type is \(\mathrm{4 + 12 = 16}\) people.

Final Answer

\(\mathrm{X@Y}\) consists of 16 integers.

This matches answer choice B. We can verify this makes sense: we started with \(\mathrm{10 + 18 = 28}\) total elements if we count everything, but we double-counted the 6 overlapping elements. The symmetric difference excludes these 6 overlapping elements entirely, leaving us with \(\mathrm{28 - 6 - 6 = 16}\) elements.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the symmetric difference operation \(\mathrm{X@Y}\)
Students often confuse \(\mathrm{X@Y}\) with the union operation (\(\mathrm{X \cup Y}\)) and think it means "all elements in either X or Y." They miss the crucial "but not both" part of the definition. This leads them to calculate the total union instead of excluding the overlapping elements.

2. Confusing "but not both" with intersection
Some students misread the operation and think \(\mathrm{X@Y}\) refers only to the elements that are in both sets (the intersection), completely opposite to what the symmetric difference actually represents.

3. Not recognizing this as a Venn diagram problem
Students may not visualize this as overlapping sets and fail to break down the problem into "only in X," "only in Y," and "in both" categories, making it harder to systematically approach the solution.

Errors while executing the approach

1. Arithmetic errors when subtracting overlapping elements
Even with the correct approach, students make simple calculation mistakes like \(\mathrm{10 - 6 = 3}\) or \(\mathrm{18 - 6 = 11}\), leading to an incorrect final sum.

2. Double-counting or incorrectly handling the overlap
Students might add the 6 overlapping elements to their final answer instead of excluding them, or subtract them incorrectly from the wrong sets.

3. Using the wrong formula application
Some students might apply \(\mathrm{|X \cup Y| = |X| + |Y| - |X \cap Y| = 10 + 18 - 6 = 22}\), which gives the union rather than the symmetric difference.

Errors while selecting the answer

1. Selecting the union instead of symmetric difference
After calculating both values during their work (22 for union and 16 for symmetric difference), students pick 22 (choice C) because they second-guess themselves about which operation was actually asked for.

2. Choosing the intersection value
Students who completely misunderstood the problem might select choice A (6), thinking the answer is simply the number of elements in both sets.