# 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}\(\times (B\cap C) = \)
- \(A \cap C =\)

annotated solution

### 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 \((A \cap B) \subseteq C\).

Hints for getting started

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Annotated solution

### Problem 3

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

Hints

Solution

### 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 \(L(a,m) \subseteq L(r,m)\).

Hints

Solution