Set theory spring2022

1. (4) List exactly four elements of each of the following sets.
a. {𝑦 | 𝑦 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑡𝑖𝑛𝑒𝑛𝑡}
b. {5𝑚 | 𝑚 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟, 𝑚 𝑖𝑠 𝑜𝑑𝑑}

2. (6) List all elements of the following sets as a set. All answers must be exact and not
rounded.
5
a. {𝑛 | 𝑛 ∈ {10, 15, 20, 25, 30}}
b. {𝑏 ∈ 𝑡ℎ𝑒 𝑎𝑙𝑝ℎ𝑎𝑏𝑒𝑡 | 𝑏 𝑝𝑟𝑒𝑐𝑒𝑑𝑒 ′𝑚′}
*note that precede means “comes before”
c. {𝑞 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 | 𝑞 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 343}

3. (6) Describe the following sets using proper set-builder notation as explained in
your book. You may not simply list the numbers.
a. {0, 3, 8, 15, 24, 35}
b. The rational numbers that are strictly between -3.5 and 3.2
c. The negative odd integers that are multiples of 3

4. (8) Let A = {a, b, c, 1, 2, 3, q, r, s}, B = {a, 1, r}, and C = {a, 3, q, x, y, z}. Which of the
following statements are true? Which are false? Explain each answer in complete
sentences. Answers without explanation will receive no points.
a. 5 ∈ 𝐴
b. 𝑧 ∈ 𝐴
c. 𝐵 ⊂ 𝐴
d. {1, 3} ∈ 𝐴
e. {1, 3} ⊂ 𝐴
f. 𝐴 ⊂ 𝐴
g. 𝐵 ⊆ 𝐵
h. ∅ ⊆ 𝐶

Number 4 continuation

5. (20) Let A, B, and C be as in #4 and let U = {1, 2, 3, 5, 7, 8, 9, a, b, c, q, r, s, f, g, x, y, z}.
Determine:
a. 𝐴 ∩ 𝐵
b. 𝐴 ∩ 𝐶
c. 𝐴 ∪ 𝐵
d. 𝐴 ∪ 𝐶
e. 𝐴 − 𝐵
f. 𝐴 − 𝐶
g. 𝐵 − 𝐴
h. 𝐶 − 𝐴
i. 𝐴𝐶
j. 𝐴 ⨁ 𝐶

Number 5 continuation

6. (8) Given that U = all students at Valencia, I represents all IT students, P represents
all part-time students, and F represents all students on financial aid, draw Venn
diagrams (make sure to use circle shapes if you are typing the answers or use tools
to draw exact circle shapes if you are handwriting the answers) illustrating this
situation and shade in the following sets:
a. Non part-time IT students that are on financial aid
Need to show answer for this one with a separate Venn diagram.
b. Part-time IT students not on financial aid
Need to show answer for this one with a separate Venn diagram.
(I will assign zero points if you do not draw separate Venn diagrams for part a and
part b.)

7. (10) Given that U = all students at UCF, C represents all Chemistry majors, G
represents all graduate students, and F represents all full-time students. Let |U| =
60,000, |C| = 1200, |F| = 32,000 and |G| = 11,500. Also assume that the number of
graduate chemistry majors is 180, 130 of which are full time, and that there are 700
full-time chemistry majors. Determine the number of students who are:
a. Full-time non-graduate Chemistry majors
b. Graduate students majoring in something other than chemistry
c. Full-time students majoring in something other than Chemistry

8. (8) Let A = {1, 2, 3} and B = {c, d, f} and let U = {1, 2, 3, 5, a, b, c, d, e, f}. List the
elements of:
a. 𝐴 × 𝐵
b. 𝐴 × 𝐴𝐶

9. (5) List all 3-elements sets in the power set of {98, p, r, q}.

10. (5) Find the binary representation of 325 using the algorithm from section 1.4.
(Must show all work and must use this method only)

11. (5) Find the binary representation of 1,714 using the algorithm from section 1.4.
(Must show all work and must use this method)

12. (5) What positive integer has this binary representation (must show all the work):
1101111100011

13. (5) Calculate the following series (your final answer will be a number) showing all
work:
6

∑(𝑘 2 − 2𝑘)
𝑘=1

14. (5) Verify the following for n = 6 (by expanding both sides and showing they are the
same using algebra)
𝑛

𝑛

𝑛

∑(𝑎𝑖 + 𝑏𝑖 ) = ∑ 𝑎𝑖 + ∑ 𝑏𝑖
𝑖=1

𝑖=1

𝑖=1

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