Sets

Let $A = \{1, 2, 3, 4, 5, 6\}$, $B = \{2, 4, 6\}$, $C = \{1, 2, 3\}$ and $D = \{7, 8, 9\}$. Determine which of the following are true, false, or meaningless.

1. $A \subset B$.
2. $B \subset A$.
3. $B \in C$.
4. $\emptyset \in A$.
5. $\emptyset \subset A$.
6. $A \lt D$.
7. $3 \in C$.
8. $3 \subset C$.
9. $\{3\} \subset C$.

Let $A = \{1, 2, 3, 4, 5, 6\}$, $B = \{2, 4, 6\}$, $C = \{1, 2, 3\}$ and $D = \{7, 8, 9\}$. If the universe is $U = \{1, 2, \ldots, 10\}$, find:

1. $A \cup B$.
2. $A \cap B$.
3. $B \cap C$.
4. $A \cap D$.
5. $\bar{B \cup C}$.
6. $A \setminus B$.
7. $(D \cap \bar C) \cup \bar{A \cap B}$.
8. $\emptyset \cup C$.
9. $\emptyset \cap C$.

Let $A = \{1,2,3,4,5\}$, $B = \{3,4,5,6,7\}$, and $C = \{2,3,5\}$.

1. Find $A \cap B$.

2. Find $A \cup B$.

3. Find $A \setminus B$.

4. Find $A \cap \overline{(B \cup C)}$.

5. Find $A \times C$.

6. Is $C \subseteq A$? Explain.

7. Is $C \subseteq B$? Explain.

Find an example of sets $A$ and $B$ such that $A\cap B = \{3, 5\}$ and $A \cup B = \{2, 3, 5, 7, 8\}$.

Find an example of sets $A$ and $B$ such that $A \subseteq B$ and $A \in B$.

Find an example of sets $A$ and $B$ such that $|A| = 4$, $|B| = 5$, and $|A \cup B| = 9$.

Let $A = \{1,2,\ldots, 10\}$. How many subsets of $A$ contain exactly one element (i.e., how many singleton subsets are there)? How many doubleton subsets (containing exactly two elements) are there?

Find an example of sets $A$ and $B$ such that $|A| = 3$, $|B| = 4$, and $|A \cup B| = 5$.

Are there sets $A$ and $B$ such that $|A| = |B|$, $|A\cup B| = 10$, and $|A\cap B| = 5$? Explain.

In a regular deck of playing cards there are 26 red cards and 12 face cards. Explain, using sets and what you have learned about cardinalities, why there are only 32 cards which are either red or a face card.