CSCI 301 'Quiz' 1

Section 1: Sets and Set Notation

  1. Write down each of the following sets using the roster method:
    1. The set of all integers with absolute value greater than 2.
    2. The set of all even positive integers less than 15
  2. True or False:
    1. \(\emptyset \in \{\emptyset\}\)
    2. \(\{\emptyset\} \subseteq \{\emptyset, \{a\}\}\)
    3. \(5 \in \{x : x \in \mathbb{Z} \text{ and } x^2 < 30\}\)
    4. \(\sqrt{2} \in \mathbb{Q}\)
  3. Express the following sets using set builder notation:
    1. All integers divisible by 3
    2. All real numbers greater than 5

Section 2: Set Operations and Cardinality

  1. Let \(X = \{1, 3, 5, 7\}\) and \(Y = \{3, 4, 5, 6\}\). Find:
    1. \(X \cup Y\)
    2. \(X \cap Y\)
    3. \(X - Y\)
    4. \(|X \cup Y|\)
  2. What is the cardinality of each of the following sets?
    1. \(\{3k : k \in \mathbb{Z^+} \text{ and } k < 4\}\)
    2. \(\{\{a\}, b, \{c, d\}\}\)
    3. \(\mathcal{P}(\{0, 1\})\)
  3. Let \(A = \{a, b, c\}\) and \(B = \{1, 2\}\). Find:
    1. \(A \times B\)
    2. \(|A \times B|\)
    3. \(B \times \emptyset\)

Section 3: Logic

  1. Let \(P =\) True, \(Q =\) False, and \(R =\) True. Determine the truth value of each of the following:
    1. \(P \land (Q \lor R)\)
    2. \((P \lor Q) \land \neg R\)
    3. \(\neg (P \land Q) \lor R\)
  2. For each of the following conditional statements, determine if it is true or false:
    1. If \(2 + 2 = 4\), then Paris is in France.
    2. If \(2 + 2 = 5\), then the sun rises in the west.
    3. If water is dry, then \(1 + 1 = 2\).
  3. Consider the sets \(U = \{1, 2, 3, 4, 5, 6, 7, 8\}\), \(A = \{1, 3, 5, 7\}\), and \(B = \{1, 2, 5, 6\}\).
    1. List the elements of \(\overline{A \cap B}\).
    2. List the elements of \((A \cup B) - (A \cap B)\).