Set theory: study problems

Problem 1

Suppose we have the following sets:

Compute the following values:

annotated solution

Problem 2

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

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

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Annotated solution

Problem 3

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



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