Venn diagram illustrating the union of sets A and B.

Let’s go through each question one by one.

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Question 1: What is the definition of the union of two mathematical sets, A and B?

The union of two sets means you combine all the elements from both sets — but you don’t repeat any element if it appears in both.

So, if an element is in set A OR in set B (or in both), it goes into the union.

Looking at the options:

- “The set of all elements that are both in set A and in set B.” → That’s the *intersection*, not union. ✘
- “The set of elements that are in either set A or set B.” → YES! This matches the definition of union. ✔
- “The set of elements that are in set A, but not in set B.” → That’s part of the difference, not union. ✘
- “The set of elements that are in set B, but not in set A.” → Also part of difference. ✘

✔ Correct answer: The set of elements that are in either set A or set B.

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Question 2: Two groups surveyed students... What could be considered the universal set?

Group A surveyed: math, science, philosophy
Group B surveyed: language, business, philosophy

The universal set is the biggest group we’re considering — everything that might be included in our discussion.

In this case, since they’re surveying students on campus, and together they covered multiple departments, the most complete set would be all students on campus — because even if some weren’t surveyed, they still belong to the overall population being studied.

Other options:

- Philosophy department only? Too small — both groups surveyed more than just philosophy. ✘
- Only students surveyed by Group A? Misses Group B’s students. ✘
- Only students surveyed by Group B? Misses Group A’s students. ✘

✔ Correct answer: The set of all students on campus

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Question 3: Use the Venn Diagram to calculate A ∪ B

We need to find all elements that are in set A OR set B (or both).

Look at the diagram:

Set A contains: a, c, h, b, d, i
Set B contains: e, g, j, b, d, i

But when taking the union, we list every unique element that appears in either circle.

So let’s list them without repeating:

From A: a, c, h, b, d, i
From B: add e, g, j (since b, d, i already counted)

Total: a, b, c, d, e, g, h, i, j

Now check the options:

- {a, c, h} → Only part of A. ✘
- {f, l, k, s, r, m, q, p} → These are outside both circles — not in A or B. ✘
- {a, b, c, d, e, g, h, i, j} → YES! Matches what we found. ✔
- {e, g, j} → Only part of B. ✘

✔ Correct answer: A ∪ B = {a, b, c, d, e, g, h, i, j}

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Final Answer:
1. The set of elements that are in either set A or set B.
2. The set of all students on campus
3. A ∪ B = {a, b, c, d, e, g, h, i, j}