Try It Now

Work these exercises to see how well you understand this material. Hints and solutions to some of these exercises are given on the second page. As the course proceeds, new notations and terminology will be introduced as they are needed. Be sure to not plug-and-chug answers to the exercises. These are to aid your understanding, and it will be difficult to pass the exam if you do not take the exercises seriously. While you should not copy others' work, feel free to discuss these exercises with others who are taking this course.

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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