COT 3100 Exam 2

Find the solution to the recurrence relation by using an iterative approach. The recurrence relation an = an - 1 + 3 with the initial condition a0 = 1

The solution for the recurrence relation is an = 3 + an − 1 = 3 + 3 + an − 2 = 2 · 3 + an − 2 = 3 · 3 + an − 3 = ⋅ ⋅ ⋅= n · 3 + an − n = n · 3 + a0 = 3n + 1

Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find the value of given function. (A ∩ B) ∪ C

{0, 2, 4, 5, 6, 7, 8, 9, 10}

Let Ai = {1, 2, 3, ..., i} for i = 1, 2, 3, .... Identify n∩(i= 1) Ai

{1}

Let Ai = {1, 2, 3, 4..., i} where i = 1, 2, 3, ... Find n∩(i= 1) Ai

{1}

Use set builder notation to give a description of each of these sets. {m, n, o, p}

{x | x is a letter of the word "monopoly" other than l and y}

Find the truth set of each of the given predicates if the domain is a set of integers. P(x): x < x^2

{x ∈ Z ∣ x < x2} = {x ∈ Z | x ≠ 0 ∧ x ≠ 1}

Find the product AB. A= | 0 −1 | | 7 2 | | −4 −3 | and B= | 4 −1 2 3 0 | | −2 0 3 4 1 |

| 2 0 -3 -4 -1 | | 24 -7 20 29 2 | | -10 4 -17 -24 -3 |

Find the value of A + B. A= | −1 0 5 6 | | −4 −3 5 −2 | and B= |−3 9 −3 4 | | 0 −2 −1 2 |

| −4 9 2 10 | | -4 -5 4 0 |

Find the truth set of each of the given predicates if the domain is a set of integers. P(x): x^2 = 2

Ø

Identify the correct steps involved in proving that the set Z+ × Z+ is countable

- Define the sets Ai := {(i, n) | n ∈ Z+}, where i belongs to Z+. We can list the sets as A1 = {(1, n) | n ∈ Z+}, A2 = {(2, n) | n ∈ Z+}, and so on. - For each i, define f(n) = (i, n), ∀n ∈ Z+. Clearly, it is a one-to-one correspondence from the set of positive integers to the set Ai. - From the definition of cardinality, each Ai is a countable set. - Thus, we can think of Z+ × Z+ as a union of countable number of countable sets A1, A2, ..., An, .... - We know that the union of countable number of countable sets is countable. Thus, Z+ × Z+ is countable.

Identify the correct steps involved in proving that the union of a countable number of countable sets is countable.

- Since empty sets do not contribute any elements to unions, we can assume that none of the sets in our given countable collection of countable sets is an empty set. If there are no sets in the collection, then the union is empty and therefore countable. - Otherwise let the countable sets be A1, A2, .... Since each set Ai is countable and nonempty, we can list its elements in a sequence as ai1, ai2, ...; again, if the set is finite, we can list its elements and then list ai1 repeatedly to assure an infinite sequence. - We can put all the elements aij into a sequence in a systematic way by listing all the elements aij in which i + j = 2 (there is only one such pair, (1, 1)), then all the elements in which i + j = 3 (there are only two such pairs, (1, 2) and (2, 1)), and so on; except that we do not list any element that we have already listed. - So, assuming that these elements are distinct, our list starts a11, a12, a21, a13, a22, a31, a14, .... (If any of these terms duplicates a previous term, then it is simply omitted.) - The result of this process will be either an infinite sequence or a finite sequence containing all the elements of the union of the sets Ai. Thus, that union is countable

Let A = | 1 1 1 3 | | 2 0 4 6 | | 1 1 3 7 | Which element of A is present in the (3, 2)th position?

1

Click and drag the given steps to their corresponding step number to prove the given statement. A ∩ Ø = Ø

1) {x | x ∈ A ∧ x ∈ ∅} 2) {x | x ∈ A ∨ F} 3) {x | F} 4) Ø

Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Match the given sets in the left to the sets in the right. 1) A − B 2) B − A 3) A ∪ B 4) A ∩ B

1) Ø 2) {f, g, h} 3) {a, b, c, d, e, f, g, h} 4) {a, b, c, d, e}

Compute each of these double sums. ∑i⁢= 02∑j⁢= 03i2j3 = _____

180

Find these terms of the sequence {an}, where an = 2(-3)^n + 5^n. a5

2,639

What are the values of these sums? ∑8j = 0(2j+1−2j) = _____

511

Find two sets A and B such that A ∈∈ B and A ⊆⊆ B.

A = ∅ and B = {∅}

Let A be the set of students who live within one mile of school and let B be the set of students who walk to classes. Describe the students in each of these sets. A ∩ B

The set of students who live within one mile of school and walk to class

Determine whether each of these sets is 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. The integers that are multiples of 10

The set is countably infinite with one-to-one correspondence n ↔ 10^n.

Identify ∞∪(i=1)Ai and ∞∩(i=1)Ai for the given Ai, where i is a positive integer. Ai = [-i, i], i.e., the set of real numbers x with -i ≤ x ≤ i.

The union is R and the intersection is [-1, 1].

Identify ∞∪(i=1)Ai and ∞∩(i=1)Ai for the given Ai, where i is a positive integer. Ai = [i, ∞), i.e., the set of real numbers x with x ≥ i.

The union is [1, ∞) and the intersection is Ø.

Prove De Morgan's law by showing that (A ⁢∪ B)' = A' ∩ B' if A and B are sets. Identify the the unknowns X, Y, Z, P, Q, and R in the given membership table.

X = 0 Y = 1 Z = 1 P = 1 Q = 0 R = 1

List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence where the nth term is the number of letters in the English word for the number n. (Note: The value of n starts from 1.) The first 10 terms are

a1 = 3 a2 = 3 a3 = 5 a4 = 4 a5 = 4 a6 = 3 a7 = 5 a8 = 5 a9 = 4 a10 = 3

A person deposits $1,000 in an account that yields 9% interest compounded annually. How much money will the account contain after 100 years?

a100 ≈ $5,529,041

Find the power set of each of the following sets, where a and b are distinct elements. {b}

{ Ø, {b} }