# Search results

1. ### Difficult simplification for Arc length integral

Homework Statement Find the length of the curve x = 3 y^{4/3}-\frac{3}{32}y^{2/3}, \quad -64\le y\le 64 Homework Equations Integral for arc length (L): L = \int_a^b \sqrt{1 + (\frac{dy}{dx})^{2}} dx The Attempt at a Solution Using symmetry of the interval and the above integral for arc...
2. ### Integral involving square root and exp

Homework Statement \int\frac{dx}{\sqrt{e^{x} + 1}} Homework Equations Using u-substitution The Attempt at a Solution Let u = \sqrt{e^{x} + 1} \Rightarrow u^{2} - 1 = e^{x} Then, du = \frac{e^{x} dx}{2\sqrt{e^{x} + 1}} \Rightarrow dx = \frac{2u du}{u^{2}-1} So...
3. ### Find a basis for the subset

Homework Statement Find a basis for the subset S = {(1, 2, 1), (2, 1, 3), (1, -4, 3)} Homework Equations The Attempt at a Solution I'm not absolutely sure I'm doing this correctly but here is my attempt: First, I put the vectors in S in the rows of a matrix (using multiple...
4. ### Number of functions from one set to another

Homework Statement How many functions are there from the set {1, 2, . . . , n}, where n is a positive integer, to the set {0, 1} a) that are one-to-one? b) that assign 0 to both 1 and n? c) that assign 1 to exactly one of the positive integers less than n? Homework Equations...
5. ### Clarification on set of assigned homework problems

Hello everyone, For homework my instructor assigned problem from the book. To show which problems to do she wrote this: 4n + 1 N={1, 2, 3...}. Does this mean problems 5, 9, 13, . . . ? Thanks for any suggestions.
6. ### Hasse diagram: minimal, least, greatest

Homework Statement Let S = {2,3,4,5} and consider the poset (S, <=) where <= is the divisibility relation. Which of the following is true? 1. 3 is a minimal element 2. 4 is a greatest element 3. 2 is a least element 4. Both 2 and 3 Homework Equations The Attempt at a Solution My answer...
7. ### Which posets are lattices

Homework Statement Could someone help with this problem? Determine which of the following posets (S, <=) are lattices. 1. A = {1, 3, 6, 9, 12} and <= is the divisibility relation. 2. B = {1, 2, 3, 4, 5} and <= is the divisibility relation. 3. C = {1, 5, 25, 100} and <= is the...
8. ### Partition of Integers with mod

Homework Statement Are the following subsets partitions of the set of integers? The set of integers divisible by 4, the set of integers equivalent to 1 mod 4, 2 mod 4, and 3 mod 4. Homework Equations The Attempt at a Solution Yes, it is a partition of the set of integers...
9. ### Digraph walk, path, circuit

Homework Statement Given the graph, determine if the following sequences form a walk, path and/or a circuit. http://img843.imageshack.us/img843/5686/digraph.png [Broken] 1. a, b, c, e 2. b, c, d, d, e, c, f 3. a, b, c, f, g, a 4. b, c, d, e Homework Equations The Attempt at a Solution 1...
10. ### Find a matrix that represents the relation

Homework Statement Find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. R on {2, 3, 4, 6, 8, 9, 12} where aRb means a|b. Homework Equations The Attempt at a Solution 1 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0...
11. ### Symmetric difference of relationships

Homework Statement 2. Let R1 = {(1,1),(1,2),(2,3),(3.4), (2,4) } and R2 = {(1,1),(2,2),(2,3),(3,3),(3,4) } be relations from {1,2,3} to {1,2,3,4} R1⨁ R2 Homework Equations The Attempt at a Solution R1⨁ R2 = {(1,2), (2,2), (2, 4), (3,3)} Is this correct?
12. ### Relationship: reflexive, symmetric, antisymmetric, transitive

Homework Statement Determine which binary relations are true, reflexive, symmetric, antisymmetric, and/or transitive. The relation R on P = {a, b, c} where R = {(a, a), (a, b), (a, c), (b, c), (c, b)} Homework Equations The Attempt at a Solution Not reflexive because there is...
13. ### Relationship: reflexive, symmetric, antisymmetric, transitive

Homework Statement Determine which binary relations are true, reflexive, symmetric, antisymmetric, and/or transitive. The relation R on all integers where aRy is |a-b|<=3 Homework Equations The Attempt at a Solution The relationship is reflexive because any number minus itself will be...
14. ### Induction proof with handshakes

Homework Statement Suppose that if there are at least 2 people in a room and each person in the room shakes hands with everyone else, but not with himself. Show that the number of handshakes is (n^2-n)/2. Make sure to show P(1), P(k) and prove P(k+1) Homework Equations The...
15. ### GCD and LCD

Homework Statement What is the greatest common divisor and least common multiple of the integers below (answer should be left in exponential form)? 2^{3}, 3^{3}, 5^{1}, 11^{2}, 13^{3} and 2^{1}3^{3}5^{2}7^{4}13^{1} Homework Equations The Attempt at a Solution This is exactly the way the...
16. ### Binary and hexadecimal expansion

Homework Statement Show that the binary expansion of a positive integer can be obtained form its hexadecimal expansion by translating each hexadecimal digit into a block of four binary digits. Homework Equations The Attempt at a Solution I know that this is true but I have no...
17. ### Big-O relationship proof

Homework Statement To establish a big - O relationship, find witnesses C and k such that |f(x)| <= C|g(x)| whenever x > k. Show that x^3 is O(x^4) but that x^4 is not O(x^3). Homework Equations If f(x) is O(g(x)) then there is a C such that f(x) <= Cg(x). The Attempt at a...
18. ### Number of comparisons in an insertion sort

Homework Statement How many comparisons does the insertion sort use to sort the list n, n-1, ......2, 1? Homework Equations The Attempt at a Solution Insertion sort compares every element with every other element in the list, but I'm unsure what this question is asking. Why does...
19. ### Subtracting cartesian products

Homework Statement Let A = {1, 2}, B = {3, 4}, C = {3} What is (A x B) - (B x C)? Homework Equations The Attempt at a Solution (A x B) - (B x C) = {(1, 3), (1, 4), (2, 3), (2, 4)} - {(3, 3), (4, 3)} = {(1, 3), (1, 4), (2, 3), (2, 4)} Since there where no elements in (B x...
20. ### 1-1, onto, both, or neither

Homework Statement Is the function one-to-one, onto, both, or neither? f: Z→Z has the rule of f(n) = 4n^3 - 1 Homework Equations The Attempt at a Solution My answer: one-to-one Is this correct?
21. ### Find a formula that generates a sequence

Homework Statement Find a formula that generates the sequence: 2/(3 x 4), -3/(4 x 5), 4/(5 x 6), -5/(6 x 7), . . . Homework Equations The Attempt at a Solution Here is what I have so far: a_n = -1(n/((n + 1)(n + 2))) Now, I'm stuck. The formula generates a negative number every time. I...
22. ### Sum of sequences

Homework Statement Find the sum of the sequence: 2, -2/3, 2/9, -2/27, 2/81, . . . Homework Equations The Attempt at a Solution I can see that the number is multiplied by -1/3, but I'm unsure of how to find the sum. Any pointers?
23. ### Floor function proof

Homework Statement Note: [x] denotes the floor of x. Prove that [3x] = [x] + [x + 1/3] + [x + 2/3] Homework Equations The Attempt at a Solution Let x = n + E, where n is an integer and 0≤ E < 1. We have three cases: Case 1: 0 ≤ E < 1/3 3x = 3n + 3E and [3x] = 3n, since 0...
24. ### Compostion of functions

Homework Statement Suppose f : R→R and g : R→R where g(x) = 3x + 5 and g o f(x) = x^2 - 7. Find the rule for f. Homework Equations The Attempt at a Solution These seem pretty simple, but I want to make sure I'm on the right track. 3(f(x)) + 5 = x^2 - 7 3(f(x)) = x^2 - 12 f(x) = (x^2 -...
25. ### Compostion of functions

Homework Statement Let g : A → B and f : B → C where A = {a,b,c,d}, B = {1,2,3}, C = {2,3,6,8}, and g and f are defined by g = {(a,2),(b,1),(c,3),(d,2)} and f = {(1,8),(2,3),(3,2)}. Find f o f Homework Equations The Attempt at a Solution I know how to find f o g by working...
26. ### One-to-one and onto

To me this problem doesn't seem right. Here it is: Is the following function one-to-one, onto, both, or neither? f: R→N f(x) = ceiling 2x/3 My answer: onto Although, wouldn't this function be invalid since it produces negative numbers and the set of natural numbers doesn't include...
27. ### One-to-one and onto

Hello all, This is tripping me up a bit an I just want to see if I on the right track. Here is the problem: Give a function from Z to N that is onto N but not one-to-one. Answer: f(x) = {x if x ≥ 0, -1x if x < 0 Seems simple, but I think it works. Note: in our book, 0 is included in...
28. ### Determine if these are functions

Hello everyone, I just want to make sure I'm doing these problems correctly. Here they are Are the following functions? 1. F : Z→Z where F(x) = 4/7x + 1 Answer: Not a function. F(1) is not an integer. 2. G : R→R where G(x) = {2x + 2 if x ≥ 0, x - 3 if x ≤ 0 Answer: Not a function...
29. ### Set operations

Here's the problem: Suppose U = {x | -10 ≤ x ≤ 10 and x ∈ Z}, A = all multiples of 2, B = all multiples of 3, and C = {-10, -9, -8, -6, -4, 0, 1, 3, 5, 6, 8, 10}. Find C - (A-B). Solution: A = {-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10} B = {-9, -6, -3, 0, 3, 6, 9} C = {-10, -9, -8, -6, -4, 0, 1...
30. ### Power Sets

While practicing power set problems I came across one that has me stumped. The problem asks: Is the following set is a power set of of a set? {∅, {b, ∅}, {a}, {a, b}, {b}} My answer: This set has 5 elements. Since 5 is not a power of 2, this cannot be the power set of any set. The...