# Set theory: study problems

### Problem 1

Suppose we have the following sets:

• A = {green, red}
• B = {3, 8}
• C = {4, 8}

Compute the following values:

• $$A \times (B\cup C) =$$
• $$B\cap C =$$
• {camel}$$\times (B\cap C) =$$
• $$A \cap C =$$

### Problem 2

Let's define sets A, B, C as follows:

• $$A = \{(p,q) \in \mathbb{Z}^2 \ \mid 3pq + 15p -5q -25 \ge 0\}$$
• $$B = \{(s,t) \in \mathbb{Z}^2 \ \mid t \ge 0\}$$
• $$C = \{(x,y) \in \mathbb{Z}^2 \ \mid x \ge 0\}$$

Prove that $$(A \cap B) \subseteq C$$.

### Problem 3

Let A, B, and C be sets. Prove that $$(A-B) \cup (B-C) \subseteq (A \cup B) - (A \cap B \cap C)$$.

### Problem 4

For any integers s and t, we'll define the set L(s,t) as follows: $$L(s,t)=\{sx+ty \mid x,y \in \mathbb{Z}\}$$

Prove the following claim. Your proof must use the definitions of congruence and divisibility directly. Do not use other lemmas that you might know about divides or congruence relationships.

Claim: For any integers a, r, m, where m is positive, if $$a \equiv r \pmod{m}$$, then $$L(a,m) \subseteq L(r,m)$$.