Let's solve the questions one by one, explaining the reasoning behind each answer.
### Question 1:
If \( A \) and \( B \) are any two sets, then \( A \cup(A \cap B) \) is equal to:
- Option a) \( A \)
- Option b) \( B \)
- Option c) \( A = B \)
- Option d) \( A \neq B \)
Solution:
Let’s analyze the expression \( A \cup(A \cap B) \):
- \( A \cap B \) is the set of elements that are both in \( A \) and \( B \).
- \( A \cup (A \cap B) \) means we take all elements that are in \( A \) or in both \( A \) and \( B \).
Intuitively, any element of \( A \cap B \) is already in \( A \). Therefore, \( A \cup (A \cap B) \) will just be \( A \), since combining \( A \) with any subset of \( A \) doesn’t add any new elements.
Thus, the correct answer is:
- \(\boxed{a)}\) \(A\)
### Question 2:
If \( A \cap B = B \), then:
- Option a) \( A \subset B \)
- Option b) \( B \subset A \)
- Option c) \( B = 0 \)
- Option d) \( B = A \)
Solution:
Given \( A \cap B = B \), for \( B \) to be identical to \( A \cap B \), all elements of \( B \) must be in \( A \). Therefore, \( B \subset A \).
Thus, the correct answer is:
- \(\boxed{b)}\) \( B \subset A \)
### Question 3:
If \( A \), \( B \), and \( C \) are any three sets, then \( A \square (B \cup C) \) is equal to:
- Option a) \( A \times B \cap A \times C \)
- Option b) \( A \cup B \times B \cup A \)
- Option c) \( A \times B \cup A \times C \)
- Option d) none of these
Solution:
This question asks for an operation \( A \square (B \cup C) \). Let's consider the properties of union:
- The distributive property tells us that \( A \cup (B \cup C) = (A \cup B) \cup (A \cup C) \), but this doesn’t directly match any options provided.
Examining each choice, none seems to match the typical set operations. Therefore, the correct option must be:
- \(\boxed{d)}\) none of these
### Question 4:
The smallest set \( A \) such that \( A \cup \{1, 2\} = \{1, 2, 3, 5, 9\} \) is:
- Option a) \(\{3,5,9\}\)
- Option b) \(\{2,3,5\}\)
- Option c) \(\{1,2,5,9\}\)
- Option d) none of these
Solution:
Given that \( A \cup \{1, 2\} \) must result in \(\{1, 2, 3, 5, 9\}\),
the elements \(1\) and \(2\) are already included from the set \(\{1, 2\}\). Therefore, the elements \(\{3, 5, 9\}\) must be provided by set \( A \).
Thus, the smallest set \( A \) would be:
- \(\boxed{a)}\) \(\{3, 5, 9\}\)
### Question 5:
Every set is a subset of:
- Option a) natural number
- Option b) itself
- Option c) equivalent set
- Option d) none of these
Solution:
By the definition of subsets, every set is always a subset of itself because all elements in the set are contained within the set itself.
Thus, the correct answer is:
- \(\boxed{b)}\) itself
By organizing our answers, we get:
1. a) \(A\)
2. b) \(B \subset A\)
3. d) none of these
4. a) \(\{3, 5, 9\}\)
5. b) itself
