Math 2366, Homework #3, Summer 2014
Name ____________________________________
Instructions: Work problems on a separate sheet of paper and attach work to this page. You should
show all work to receive full credit for problems. Checking your work with computer algebra systems is
fine, but that doesn’t count as “work” since you won’t be able to use CAS programs on exams or quizzes.
Sketch any graphs you obtain. Questions with compact answers should be recorded directly on this
page. Graphs and longer answers that won’t fit here, indicate which page of the work the answer can
be found on and be sure to clearly indicate it on the attached pages. Some of the problems will be
proofs. These will require explanations and not just computations.
1. Why is f not a function from R to R?
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a. 𝑓(𝑥) = 𝑥
b. 𝑓(𝑥) = √𝑥
c. 𝑓(𝑥) = ±√𝑥 2 + 1
2. Find the domain and range of these functions.
a. The function that assigns to each nonnegative integer its last digit.
b. The function that assigns to a bit string the number of bits in the string
c. The function that assigns to each positive integer the largest perfect square not exceeding
this integer.
d. The function that assigns to each positive integer its largest decimal digit.
e. The function that assigns to each pair of positive integers, the maximum of these integers.
3. Determine whether the functions are one-to-one? Onto? A bijection? For the specified domain
and range.
a. 𝑓(𝑛) = 𝑛2 + 1, 𝑍 → 𝑍
b. 𝑓(𝑛) = |𝑚| − |𝑛|, 𝑍 × 𝑍 → 𝑍
c. The function that assigns to each student their final grade in the class.
𝑥+1
d. 𝑓(𝑥) = 𝑥+2 , 𝑅 → 𝑅
e. 𝑓(𝑥) = 𝑥 5 − 1, 𝑅 → 𝑅
4. Give an example of a function from N→N that is one-to-one, but not onto.
5. Draw the graph of the functions.
𝑥
𝑥
a. 𝑓(𝑥) = ⌈ ⌉ ⌊ ⌋
2 2
𝑥
1
b. 𝑓(𝑥) = ⌊2 ⌈2⌉ + 2⌋
6. Find the first ten terms of the sequence. Write an expression for the sequence (either recursive
or explicit).
a. The sequence that begins with 2 and in which each successive term is 3 more than the
preceding terms.
b. The sequence that lists the odd positive integers in increasing order, listing each odd integer
twice.
c. The sequence whose first term is 2 and the second is 4, and whose each succeeding term is
the sum of the previous two terms.
d. The sequence whose terms are the difference of the factorial and the corresponding power
of two starting from n=0.
e. The sequence whose nth term is the largest integer k such that 𝑘! ≤ 𝑛.
7. Find the first six terms of the sequence.
a. 𝑎𝑛 = 𝑎𝑛−1 + 3𝑎𝑛−2 , 𝑎0 = 1, 𝑎1 = 2
b. 𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−3 , 𝑎0 = 1, 𝑎1 = 2, 𝑎2 = 0
c. 𝑎𝑛 = 𝑛 + (−1)𝑛
d. 𝑎𝑛 = 𝑎𝑛−1 + 2𝑎𝑛−2 + 2𝑛 + 9, 𝑎0 = 1, 𝑎1 = 0
8. Find a formula for the sequence, and determine the next three terms.
a. 1,0,1,1,0,0,1,1,1,0,0,0,1,…
b. 15, 8, 1, -6, -13, -20, -27,…
c. 2, 3, 7, 25, 121, 721, 5041, 40321,…
d. 0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, …
e. 2, 4, 16, 256, 65536, 4294967296, …
9. Find the value of each sum.
a. ∑4𝑗=0(−2)𝑗
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b. ∑𝐽∈𝑆 (𝑗 ) , 𝑆 = {1,3,5,7}
c. ∑2𝑖=1 ∑3𝑗=1(𝑖 + 𝑗)
d. ∑3𝑖=1 ∑4𝑗=1(𝑖 2 𝑗)
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e. ∑𝑛𝑘=1 𝑘(𝑘+1)
f.
∑𝑛𝑘=1(2𝑘 − 1)
10. Determine whether each of these sets in finite, countably infinite, or uncountable. For those
that are countably infinite, exhibit a one-to-one correspondence between the set of positive
integers and that set.
a. The negative integers
b. The integers less than 100
c. The integers that are multiples of 7
d. The odd negative integers
e. The set 𝐴 × 𝑍 + where A={2,3}
f. All positive rational numbers than cannot be written with the denominators less than 4
g. Integers not divisible by 3
h. The real numbers with the decimal representation consisting of all 1s.
i. 𝑍 + × 𝑍 +
11. Give an example of two uncountable sets A and B such that A—B is a) finite, b) countably
infinite, c) uncountable.
12. Let 𝑃(𝑛) be the statement that 12 + 22 + 32 + ⋯ + 𝑛2 =
n. Prove this by mathematical induction.
𝑛(𝑛+1)(2𝑛+1)
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for any positive integer
13. Prove that is 𝐴1 , 𝐴2 , … , 𝐴𝑛 and 𝐵1 , 𝐵2 , … , 𝐵𝑛 are sets such that 𝐴𝑗 𝐵𝑗 for j=1,2,…,n, then
⋃𝑛𝑗=1 𝐴𝑗  ⋃𝑛𝑗=1 𝐵𝑗 .
14. Find 𝑓(1), 𝑓(2), 𝑓(3), 𝑓(4) if 𝑓(𝑛) is defined recursively by 𝑓(0) = 1 and
a. 𝑓(𝑛 + 1) = 𝑓(𝑛) + 2
b. 𝑓(𝑛 + 1) = 2 𝑓(𝑛)
c. 𝑓(𝑛 + 1) = 𝑓(𝑛)2 − 2𝑓(𝑛) − 2
d. 𝑓(𝑛 + 1) = 𝑓(𝑛 − 1)/𝑓(𝑛); assume 𝑓(1) = 2
15. Give a recursive definition of
a. The set of odd positive integers
b. The set of polynomials with integer coefficients
c. The set of positive integers divisible by 5
d. The set of integers one less than integers divisible by three
16. Show that the proposed recursive definition of a function on the set of positive integers does
not produce a well-defined function.
𝑛
a. 𝐹(𝑛) = 1 + 𝐹 (⌊2 ⌋), for 𝑛 ≥ 1 and 𝐹(1) = 1.
𝑛
b. 𝐹(𝑛) = 1 + 𝐹 (2 ) if n is even and 𝑛 ≥ 2, 𝐹(𝑛) = 𝐹(3𝑛 − 1) if n is odd and 𝑛 ≥ 3, and
𝐹(1) = 1.