Exercises

Precalculus for Everybody

Chapter 1. Preliminaries

An Overview of Logic and Set Theory
  1. If \(\mathit{U} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}\), \(A = \{x \in \mathit{U} / x \mbox{ is even}\}\), \(B = \{x / x \mbox{ is prime}\}\)   and   \(D = \{ x \in \mathit{U} / x \mbox{ is not a factor of 9} \}\), find:
    1. \[ \complement (A \cup B) \]
    2. \[ A - \complement D \]
    3. \[ B - \complement D \]
    4. \[ (A-B) \cup (B-A) \]
  2. If \(\mathit{U} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \),   find the sets \(A\)   and   \(B\):
    1. \[ A - B = \{7, 9\} \]
    2. \[ \complement A \cap \complement B = \{ 6 \} \]
    3. \[ A \cap B = \{ 0, 5, 8 \} \]
  3. If \(\mathit{U} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\),   find the sets \(A\)   and   \(B\):
    1. \[ (A - B) \cup (B - A) = \{0, 1, 5, 8\} \]
      \[ \begin{aligned} &(A - B) \cup (B - A) \\[1em] &\hspace{5em}= \{0, 1, 5, 8\} \end{aligned} \]
    2. \[ \complement A \cup \complement B = \{ 0, 1, 2, 5, 6, 7, 8 \} \]
    3. \[ A \cap \complement B = \{ 0, 8\} \]
  4. Find the sets \(A\), \(B\), \(C\)   and   \(U\), the universal set, if:
    1. \[ \complement A = \{1, 4, 7, 8, 9\} \]
    2. \[ \complement B = \{2, 4, 5, 7\} \]
    3. \[ \complement C = \{2, 4, 7, 8\} \]
    4. \[ A \cap B \cap C = \{0, 3, 6\} \]
    1. \(\{ 1, 9\}\)
    2. \(A=\{ 2,4,6,8 \}\)
    3. \(\{ 2,5,7 \}\)
    4. \(\{ 3,4,5,6,7 \}\)
  2. \[ A = \{ 0,5,7,8,9 \}, \quad B = \{ 0,1,2,3,4,5,8 \} \]
    \[ \begin{aligned} &A = \{ 0,5,7,8,9 \}, \\[1em] &B = \{ 0,1,2,3,4,5,8 \} \end{aligned} \]
     
  3. \[ A = \{ 0,3,4,8,9 \}, \quad B = \{ 1,3,5,6,9 \} \]
    \[ \begin{aligned} &A = \{ 0,3,4,8,9 \}, \\[1em] &B = \{ 1,3,5,6,9 \} \end{aligned} \]
     
  4. \[ A = \{ 0, 2 , 3 , 6 \} , \quad B = \{ 0, 1, 2, 6, 8, 9 \}, \quad C = \{ 0, 1, 3, 5, 6, 9 \}, \quad U = \{ 0, 1, 3, 4, 5, 6, 8, 9 \} \]
    \[ \begin{aligned} &A = \{ 0, 2 , 3 , 6 \} , \\[1em] &B = \{ 0, 1, 2, 6, 8, 9 \}, \\[1em] &C = \{ 0, 1, 3, 5, 6, 9 \}, \\[1em] &U = \{ 0, 1, 3, 4, 5, 6, 8, 9 \} \end{aligned} \]