COT 3100 Homework 7

Which of the following rules generates a sequence beginning with the terms 1, 2, 4?

- Each term is twice the previous term. - The nth term is obtained by adding increasing values to the previous term. - The terms are positive integers that are not multiples of 3.

Identify the correct steps involved in proving that if A, B, and C are sets such that |A| ≤ |B| and |B| ≤ |C|, then |A| ≤ |C|.

- The definition of |A| ≤ |B| is that there is a one-to-one function f: A → B. Similarly, we are given a one-to-one function g: B → C. - We need to show that g ◦ f is one-to-one. This means that we need to show that if x and y are two distinct elements of A, then g(f(x)) ≠ g(f(y)). - First, since f is one-to-one, the definition tells us that f(x) ≠ f(y). - Second, since now f(x) and f(y) are distinct elements of B, and since g is one-to-one, we conclude that g(f(x)) ≠ g(f(y)), as desired. - Therefore, by definition, |A| ≤ |C|.

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 odd negative integers.

- The set is countably infinite. - The set is countably infinite with one-to-one correspondence 1 ↔ -1, 2 ↔ -3, 3 ↔ -5, 4 ↔ -7, and so on. - The one-to-one correspondence is given by n ↔ -(2n - 1).

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 greater than 10.

- The set is countably infinite. - The set is countably infinite with one-to-one correspondence 1 ↔ 11, 2 ↔ 12, 3 ↔ 13, and so on. - The one-to-one correspondence is given by n ↔ (n + 10).

Find these terms of the sequence {an}, where an = 2(-3)^n + 5^n. a1

-1

What are the values of these sums? ∑5/k= 1(k+ 1)∑k⁢= 15(k⁢+ 1) = _____

20

Find these terms of the sequence {an}, where an = 2(-3)^n + 5^n. a0

3

Let A = | 1113 | | 2046 | | 1137 | What size is A?

3 x 4

Let A be a 3 × 4 matrix, B be a 4 × 5 matrix, and C be a 4 × 4 matrix. Determine the products that are defined, and find the size of the products that are defined. AB

AB is defined, and its dimension is 3 × 5.

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 real numbers between 0 and 2

The set is uncountable.

Find the solution to the recurrence relation by using an iterative approach. The recurrence relation an = -an - 1 with the initial condition a0 = 5

The solution for the recurrence relation is an = −an − 1 = (−1)2an−2 = ⋅ ⋅ ⋅ = (−1)nan − n = (−1)na0 = 5 ⋅ (−1)n

Let A= | −1 2 | | 1 3 | Identify the steps to prove that (A^-1)^3 is the inverse of A^3.

The value of A^3= | 1 18 | | 9 37 | and (A^−1)^3= | -37/125 18/125 | | 9/125 -1/125 | ---- The value of (A^3)^−1= | 1 18 | ^-1 | 9 37 | = | -1/125 18/125 | | 9/125 -37/125 | =(A^−1)^3

Let A be a 3 × 4 matrix, B be a 4 × 5 matrix, and C be a 4 × 4 matrix. Determine the products that are defined, and find the size of the products that are defined. BA

This is not defined since the number of columns of B does not equal the number of rows of A.

If it is possible to label each element of an infinite set S with a finite string of keyboard characters, from a finite list characters, where no two elements of S have the same label, then S is a countably infinite set. Use the above statement and prove that the set of rational numbers is countable.

We can label the rational numbers with strings from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, /, −} by writing down the string that represents that rational number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.

Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions. For the sequence an = −2an − 1 and a0 = −1, the values of the first six terms are:

a0 = -1 a1 = 2 a2 = -4 a3 = 8 a4 = -16 a5 = 32

What are the terms a0, a1, a2, and a3 of the sequence {an}? For the sequence an = (n + 1)^(n + 1), the values of the first four terms of the sequence are:

a0 = 1 a1 = 4 a2 = 27 a3 = 256

List the first 10 terms of each of these sequences. Do not enter commas for numbers greater than 1000. The sequence that begins with 2 and in which each successive term is 3 more than the preceding term. The first 10 terms are:

a1 = 2 a2 = 5 a3 = 8 a4 = 11 a5 = 14 a6 = 17 a7 = 20 a8 = 23 a9 = 26 a10 = 29

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

A person deposits $1,000 in an account that yields 9% interest compounded annually. Set up a recurrence relation for the amount in the account at the end of n years.

an = 1.09 * an - 1

Find the product AB. A= | 1 0 1 | | 0 −1 −1 | | −1 1 0 | and B= | 0 1 −1 | | 1 −1 0 | | −1 0 1 |

| -1 1 0 | | 0 1 -1 | | 1 -2 1 |

Let A= | −1 2 | | 1 3 | Find A^-1.

| -3/5 2/5 | | 1/5 1/5 |

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

| 0 3 9 | | 1 4 -1 | | 2 -5 -3 |

Let A= | −1 2 | | 1 3 | Find A^3.

| 1 18 | | 9 37 |

Let A = | 1113 | | 2046 | | 1137 | Identify the third column of A.

| 1 | | 4 | | 3 |

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

| 4 -1 -7 6 | | -7 -5 8 5 | | 4 0 7 3 |