Homework Statement
I need to prove that different insertion orders of the same keys always gives us a different binary tree.
Homework Equations
All obvious BST properties apply:
The left subtree of a node contains only nodes with keys less than the node's key.
The right subtree of a...
I understand that the nth harmonic number grows about as fast as the natural logarithm of n, because the sum of ln(n) is integral of 1/x between 1 and n.
What I don't understand is what makes ln(n) become the upper/lower bound of the nth harmonic number ? What makes it bigger or smaller, so it...
Homework Statement
Ok, basically I need to show that Ʃ 1/n (between 1 and n) (which is harmonic number) is θ (big theta) of ln(n), which means that is it bounded below and above by this function(upper and lower bound). But I don't quite understand how to prove it.Homework Equations
I know...
Basically I need to find a mistake in this "proof".
I claim that 0,1,2,3...are all even.
I will use induction to prove that 'n is even' for n = 0,1,2,3...
Base case is n = 0, which is true, 0 is even. I assume that the statement is true for
n = 0,1,2,3...,k and consider n = k+1. By...
a little bit more explanation with sample solution will be helpful. What you said doesn't help a bit. I know that it is an arithmetic formula (4n-1). The sum on the other hand is not.
I'm a little stuck here...
I need to write this in the summation notation, and then find and prove a formula in terms of n, using induction :3+7+11+...+(4n-1)
I know that the summation notation is
n
+---
\
/ 4i-1
+---
i=1
but I have no idea how to...
I'm a little stuck with these bad boys:
Let P(x) be the assertion “x is odd”, and let Q(x) be the assertion “x is twice an integer.” Determine whether the following statements are true:
1. (Vx ∈ Z)(P(x) ⇒ Q(x))
2. (Vx ∈ Z)(P(x)) ⇒ (Vx ∈ Z)(Q(x))
My attempt:
I don't get the statement at all...
URGENT Field Proofs help.
I need to prove the following:
1) Prove that if x, y are elements of a field, and X x Y = 0 then either x = 0 or y = 0 .
Write a detailed solution. and mention which of the eld axioms you are using.
2) Let F be a field in which 1 + 1 = 0 . Prove that for any...
I've given a function where f: R->R, and need to determine an image(or range):
f(x) = x/(1+|x|)
I've pretty sure the image is R, but I'm not positive:
Heres my attempt:
y/1 = x/(1+|x|)
y(1+|x|) = x
y+ y|x| = x
y|x| = x - y... I'm kinda stuck here, since I can't determine an image...
I need to graph/find numbers for S∩T where S is x^2+y^2 <=100 and T is x+y<=14.
I know I can find them simply by choosing/picking them, but are there any other solution ?
I thought maybe doing
x^2+y^2 <=100
+
x+y<=14
=
x^2+y^2 + x+y<=14 +100 =
x^2+y^2 + x+y<=114 =
x^2+y^2 <=...
Ok, I think I'm getting closer:
Here's what I've got:
(x-z)^2 - 4(yx - yz + y^2 + 2y) >=0
now I get why y = 0, would give me (x-z)^2...
but how would y be equal to 0 ?
I don't get this clue :tongue2: How's y ever going to be 0 ?
Even if y is 0, it will be (x+z)^2 >= 4xz, which isn't very helpful...
I need to prove this:
(x-2y+z)^2 >= 4xz -8y.
using this:
x+z<=2:I can't seem to simplify (x-2y+z)^2 >= 4xz -8y enough to get x,y,z by themselves..
I expanded...
Question:
I need to prove this inequality:
Where x,y,x are non-negative and x+z<=2:
(x-2y+z)^2 >= 4xz -8y.
My attempt:
I thought maybe choosing x as 0 and z as 0 will and then solving for y... but that only yields y+2 >= 0, which isn't really a solution, since I can't choose numbers...