CSCI 301 L15 Worksheet

Lecture 15 - Exercises

Part A - Modular Congruence

  1. True or false:
    1. \(9 \equiv 1 \pmod{4}\)
    2. \(10 \equiv 20 \pmod{2}\)
    3. \(20 \equiv 4 \pmod{8}\)
    4. \(22 \equiv 8 \pmod{6}\)
    5. \(\forall x \in \mathbb{Z}, n \in \mathbb{N}, x \equiv x \mod n\)
    6. \(\forall n \in \mathbb{N}, 3n \equiv 0 \pmod{n}\)
  2. Prove that if \(a\) and \(b\) have the same remainder when divided by \(n\), then \(a \equiv b \pmod{n}\).

Part B - Proofs with Sets

  1. Prove that \(\{12n : n \in \mathbb{Z}\} \subseteq \{2n: n \in \mathbb{Z}\} \cap \{3n: n \in \mathbb{Z}\}\).
  2. Suppose \(A, B,\) and \(C\) are sets. Prove that if \(B \subseteq C\), then \(A \times B \subseteq A \times C\).
  3. Supoose \(A, B,\) and \(C\) are sets. Prove that \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\).
  4. Prove that if \(A\) and \(B\) are both sets with universal set \(U\), then \(\overline{A \cap B} = \overline{A} \cup \overline{B}\).