Let U
be a set (the "universe"), and A,B⊆U
. Consider the following two statements:
(i) |U−A|=|A′|

(ii) |A∪B|=|A|+|B|
Both statements are true
Statement (i) is true; (ii) is false
Statement (i) is false; (ii) is true
Both statements are false

Let's analyze the two statements to determine their validity:

### Statement (i): \(|U - A| = |A'|\)

Here:
- \( U - A \) refers to the set difference, which means all elements in \(U\) that are not in \(A\).
- \( A' \) (also denoted as the complement of \(A\)) refers to all elements in \(U\) that are not in \(A\).

Since both \( U - A \) and \( A' \) refer to the same set, this means \(|U - A| = |A'|\) is true. Both sets contain the same elements, and hence, their cardinalities are equal.

**Conclusion**: Statement (i) is **true**.

---

### Statement (ii): \(|A \cup B| = |A| + |B|\)

This statement says that the cardinality of the union of sets \(A\) and \(B\) is equal to the sum of the cardinalities of \(A\) and \(B\).

This is **false** in general unless \(A\) and \(B\) are disjoint sets (i.e., \(A \cap B = \emptyset\)).

In most cases, \(A\) and \(B\) may have common elements, and the correct formula should account for this overlap:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
So, if \(A\) and \(B\) overlap, the union will not simply be the sum of their cardinalities.

**Conclusion**: Statement (ii) is **false**.

---

### Final Answer:
- **Statement (i) is true**, and **statement (ii) is false**.

Thus, the correct answer is: **Statement (i) is true; (ii) is false**.

---

Would you like further details or clarification on any of the points?

Here are 5 related questions to expand your understanding:
1. What is the relationship between set difference and set complement?
2. How does the inclusion-exclusion principle apply to unions of multiple sets?
3. When are two sets disjoint, and how does this affect their union?
4. Can you provide an example where the formula for \(|A \cup B|\) differs from \(|A| + |B|\)?
5. What are the implications of these set properties in probability theory?

**Tip:** Always be cautious when dealing with set unions—account for any overlap using the inclusion-exclusion principle.

Let U be a set (the 'universe'), and A,B⊆U. Consider the following two statements: (i) |U−A|=|A′| (ii) |A∪B|=|A|+|B|. Both statements are true; Statement (i) is true; (ii) is false; Statement (i) is false; (ii) is true; Both statements are false.

Analyze two statements in set theory: (i) |U − A| = |A'| and (ii) |A ∪ B| = |A| + |B|. Learn how the inclusion-exclusion principle applies to unions and how set complement relates to set difference. Suitable for students in Grades 10-12.

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