Exercises

  1. 1. List four elements of each of the following sets:
    1. \(\left \{ k \in \mathbb{P} \; | \; k - 1 \; \mathrm{is\; a \; multiple\; of\; } 7 \right \}\)
    2. \(\left \{ x \; | \; x \; \mathrm{is\; a\; fruit\; and\; its\; skin\; is\; normally\; eaten} \right \}\)
    3. \(\left \{ x \in \mathbb{Q} \; | \; \frac{1}{x} \in \mathbb{Z} \right \}\)
    4. \( \left \{ 2n \; | \; n \in \mathbb{Z} , n < 0 \right \} \)
    5. \(\left \{ s \; | \; s = 1 + 2+ . . . + n \mathrm{\; for \; some\; n \in \mathbb{P}} \right \}\)

  2. List all elements of the following sets:
    1. \(\left \{ \frac{1}{n} \; | \; n \in \left \{ 3,4,5,6 \right \} \right \}\)
    2. \(\left \{ \alpha \in \mathrm{\; the \; alphabet \;} | \; \alpha \mathrm{\; precedes \; F} \right \}\)
    3. \(\left \{ x \in \mathbb{Z} \; | \; x=x+1 \right \}\)
    4. \(\left \{ n^2 \; | \; n = -2,-1,0,1,2 \right \}\)
    5. \(\left \{ n \in \mathbb{P} \; | \; n \; \mathrm{is \; a \; factor \; of \;} 24 \right \}\)

  3. Describe the following sets using set-builder notation.
    1. {5, 7, 9, . . ., 77, 79}
    2. the rational numbers that are strictly between -1 and 1
    3. the even integers
    4. {-18, -9, 0, 9, 18, 27, . . .}

  4. Use set-builder notation to describe the following sets:
    1. {1, 2, 3, 4, 5, 6, 7}
    2. {1, 10, 100, 1000, 10000}
    3. {1, 1/2, 1/3, 1/4, 1/5}
    4. {0}

  5. Let A = {0, 2, 3}, B= {2, 3}, and C = {1, 5, 9}. Determine which of the following statements are true. Give reasons for your answers.
    1. \( 3 \in A\)
    2. \(\left \{ 3 \right \} \in A\)
    3. \(\left \{ 3 \right \} \subseteq A\)
    4. \( B \subseteq A\)
    5. \( A \subseteq B\)
    6. \(\emptyset \subseteq C\)
    7. \(\emptyset \in A\)
    8. \(A \subseteq A\)

  6. One reason that we left the definition of a set vague is Russell's Paradox. Many mathematics and logic books contain an account of this paradox. Find one such reference and read it.

 

Source: Al Doerr and Ken Levasseur, http://faculty.uml.edu/klevasseur/ads-latex/ads.pdf
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