For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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1answer
30 views

Proof with Cartesian coordinates.

Let $S_b := \{(x,y) \in\mathbb R^2 | y = 3x + b\}$ where $b\in\mathbb R$. Give a direct proof that if $(r,s)\in\mathbb R^2$, then there exists a $b\in\mathbb R$ such that $(r,s) \in S_b$. I have ...
1
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1answer
25 views
0
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0answers
22 views

Using a direct proof to prove circumscribed shapes.

I am looking at this problem: Use a Direct proof to show that if A is a circle circumscribed by a square B, and the square B is circumscribed by a Circle C, then the area of Circle C is twice the ...
2
votes
5answers
2k views

Is it true that $n^2+3n+13$ is prime for all $n\in\mathbb ℤ^+$?

Prove or disprove the statement: If $n\in\mathbb ℤ^+$, then $n^2+3n+13$ is prime. I am lost here. All I know is that $n$ is greater than or equal to one, since it is a positive integer.
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0answers
20 views

Proof for the number of leaves for any Binary Search Tree

A property for binary trees is that the number of leaves is the number of full nodes plus 1, in other words, $L = F + 1$ where $L$ is the number of leaves and $F$ is the number of full nodes. What ...
0
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3answers
117 views

How would I show that Tn=3^n + 2 is a solution to the recurrence?

Would anyone be able to help me or give me some advice on the following problem: Consider the recurrence with $T_0 = 3$ and $T_{n+1} = 3T_n - 4$ for all $n \in \mathbb{N}$. How would I show that ...
0
votes
1answer
19 views

Boolean algebra proof (a+b) (a+c)' = a'bc'

I have to prove that (a+b) (a+c)' = a'bc' My algebra skills are really rusty and I was wondering what identities are used to solve this so I can get a better understanding
-1
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2answers
41 views

Proving a matrix is always symmetric [duplicate]

$B$ is a square matrix of real numbers. Show that the matrix $BB^T$ is always symmetric.
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2answers
68 views

Prove using the definition of a limit, that $f(x) >$ something if $|x| < \delta$

The function $f (x)$ is defined for $−∞ < x < ∞$. In addition, we have $$\lim_{x \to 0} f(x) = 2$$ (a) Give the $\epsilon$-$δ$-definition of $\lim_{x \to 0} f(x) = 2$. (b) Prove (using this ...
2
votes
2answers
85 views

Suppose A and B are sets. Prove that A ⊆ B if and only if A ∩ B = A.

Here's how I see it being proved. If A and B are sets,and the intersection of A and B is equal to A, then the elements in A are in both the set A and B. Therefore, the set of A is a subset of B since ...
0
votes
3answers
72 views

Prove by induction that… $1+3+5+7+…+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$

I'm not too sure exactly how to approach this question. Would anyone be able to give me any helpful advice or some sort of direction? I have a little problem with induction. Prove by induction that: ...
2
votes
3answers
80 views

Proof: For all integers $x$ and $y$, if $x^2+ y^2= 0$ then $x =0$ and $y =0$

I need help proving the following statement: For all integers $x$ and $y$, if $x^2+ y^2= 0$ then $x =0$ and $y =0$ The statement is true, I just need to know the thought process, or a lead in the ...
2
votes
4answers
92 views

How to prove $C$ from $A \leftrightarrow (B \leftrightarrow C)$ and $A \leftrightarrow B$?

How does one prove $C$ from the premises: $A \leftrightarrow (B \leftrightarrow C)$ and $A \leftrightarrow B$ ? I've tried to prove $C$ by contradiction, using a sub-proof which presumes $\neg ...
1
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0answers
84 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
1
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1answer
27 views

Verify that $(I−XY)^{(-1)}*X=X*(I−YX)^{(-1)}$ [duplicate]

Verify that $(I_n−XY)^{-1}\cdot X=X\cdot (I_m−YX)^{-1}$ The first $I$ is of order $n$ and the second is of order $m$. $X$ is $n\times m$ $Y$ is $m\times n$
4
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4answers
303 views

How to show that these two lines are perpendicular?

Let $AEE'$ be an isoceles triangle with $\angle EAE'=90^\circ$ such that $AE=AE'$ and such that $A$, $E$ and $E'$ lie on the circle $c_1$. Let $ADD'$ be an isoceles triangle with $\angle ...
2
votes
1answer
92 views

Measuring Unsigned Simple Functions

I was hoping that someone would be able to help me solve this problem regarding simple functions and their measure. This problem is coming straight from Introduction to Measure Theory by Terrence Tao. ...
10
votes
1answer
121 views

Prove that $a < b\sqrt{3}$ under conditions given

There are integers $a$ and $b$ such that: 1) $a > b > 1$ 2) $ab+1$ is divisible by $a+b$ and $ab-1$ is divisible by $a-b$. Prove that $a < b\sqrt{3}$. It's really hard, do you see a ...
1
vote
1answer
178 views

Linear algebra proof regarding matrices

I'd like a hint rather than a full solution. The problem I am considering is the following: $X$ is an $n\times m$ matrix $Y$ is $m\times n$ Show that $(I - XY)^{-1}\cdot X = X\cdot(I - ...
1
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1answer
25 views

Boolean Algebra: making a proof assistance

So far i've tried all the identities my teacher gave us and keep getting stuck I have to prove that x'y' + y = x' + xy using boolean algebra identities
3
votes
1answer
54 views

Let a,b,c be integers. Prove that if a|c and b|c, then either a|b or b|a.

Let a,b,c be integers. Prove that if a|c and b|c, then either a|b or b|a. Any ideas? (Suggested proof by contradiction). Not really sure how to go about this.
0
votes
2answers
28 views

Call a subset A ⊆ ℝ left-infinite if either A = ℝ or A = (a, ∞) for some a ∈ ℝ.

So, as part of some extra credit my professor gave me, I am given this problem: Call a subset $A \subseteq \mathbb R$ left-infinite if either $A =\mathbb R$ or $A=(a,\infty)$ for some $a\in\mathbb ...
3
votes
1answer
29 views

Show by committee selection argument

First post in Stack Exchange and feel bad to be in need of help. But, I'm having a hard time understanding this one or rather showing the argument. $\binom{n}{k} = \binom{n-2}{k-2} + ...
0
votes
1answer
43 views

Proving a sequence is increasing and converging as $n\to \infty$.

Suppose that $x_0 \in (-1,0)$ and $x_n=\sqrt{x_{n-1}+1}-1$ for $n \in \mathbb N$. Prove that $x_n \uparrow 0$ as $n\to \infty$. What happens when $x_0 \in [-1,0]$? Before this, the problems I did had ...
0
votes
2answers
31 views

Is this the correct way to prove by induction?

Prove by induction that $$1 + 3 + 5 + 7 + ... + (2n + 1) = (n+ 1)^2 $$ //for every n ∈ $\mathbb N$. $$1+2+3+...+n=\frac{n(n+1)}2$$ Proof: $$3+5+7+\ldots+(2n+1)=$$ ...
0
votes
0answers
25 views

If $a,b,c$ are the vertices of a triangle in the complex plane, prove that the area of a triangle is $\frac{1}{2}|b-c|^2|Im\frac{c-a}{c-b}|$

I have trouble with this proof. I can get as far as the fact that we must position the vertex $c$ on the origin and then rotate by a factor of $|b-c|$. But then this gives: \begin{align*} ...
0
votes
1answer
151 views

Double Complement of a set proof

Question states: Prove the law of double complements for sets: If $A$ is a set and $A^\complement$ is its complement than prove that: $$ (A^\complement)^\complement = A$$ I started with: $$ ...
2
votes
1answer
36 views

Formal proof structure for $\forall n \in \mathbb{N}, P(n) \rightarrow \forall n \in \mathbb{N}, Q(n)$

I'm used to proving universal quantification claims (i.e. $\forall n \in \mathbb{N}, [P(n) \rightarrow Q(n)]$) by: Assuming an arbitrary number in the naturals, assuming the antecdent $P(n)$, doing ...
3
votes
2answers
70 views

If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?

The title pretty much says it all: If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true? Edit: Let me attempt to be a little more precise: ...
1
vote
2answers
66 views

how to prove $\sum_{i=1}^n i^k =\Theta(n^{k+1})$

we can say that if all $i$ s in the sum were equal to $n$ then the answer to the summation would be $n\cdot n^k$. So $n^{k+1}$ is the upper bound.so $\displaystyle\sum_{i=1}^n i^k=O(n^{k+1})$ For ...
1
vote
1answer
51 views

Determining if two statements are equivalent, logical sense.

I am confused, I am working with proofs and I have the following statement to work with $\forall n\in\mathbb{N},P(n) \implies P(n+1)$ I have a second statement $\forall n\in\mathbb{N}, ...
2
votes
4answers
57 views

Prove $\frac{n}{n+1}<\frac{n+1}{n+2}$

How can we prove the following inequality: $$\frac{n}{n+1} < \frac{n+1}{n+2}$$ I understand how to do proof by inductions and contradictions, but I am getting stuck on this question. My proof ...
1
vote
1answer
33 views

Suppose n is an integer. Use a proof by contrapositive to show if n^3 is even, then n is even

So, we assume that n is not even. Then, $n$ is odd, so $n= 2k+1$ for some integer $k$. Then, $(2k+1)^3 = 8x^3+12k^2+6k+1$. Would it be legal, then, for me to say $(8k^3+12k^2+6k)+1 = ...
-1
votes
2answers
32 views

Suppose that x is an integer. Use a proof by contrapositive to prove that if 5x+7 is even, then x is odd.

I know that we assume x is even. So, as x is even, x = 2k for some integer k. Then, that would make for 5(2k)+7 = 10k + 7. And this is where I'm stuck. I know that it isn't complete at 10k+7 to ...
2
votes
1answer
58 views

Statistics - Show that $\hat{\theta}$ hat is a biased estimator of $\theta$

I'm asked to solve this exercise, but I can't manage to find something satisfying. Any help/hint would be much appreciated. Let $Y_1, Y_2,\dots, Y_n$ denote a random variable sample of size n from a ...
0
votes
1answer
37 views

Characterization of analytic functions by exponential functions

Let $f$ be an analytic function on domain $D$ such that $f(z) \neq 0, \forall z \in D.$ Could anyone advise me how to prove $f= e^{h},$ for some analytic $h$ on $D \ ?$ Thank you.
0
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3answers
67 views

Proofs about Matrix Rank

I'm trying to prove the following two statements. I can prove them easily by considering the matrix as a representation of a linear map with a given basis, but I don't know a proof which uses just the ...
0
votes
1answer
20 views

Help understanding proof for: Let $X$ be a set. Then $X \not\approx P(X)$ (where $\approx$ is equivalence relation)

In trying to understand the following proof, I am getting stuck on the chosen definition of $Y = \{ x \in X \mid x \not\in f(x) \}$. How do we know that such a set exists in $P(X)$ when we don't even ...
3
votes
2answers
36 views

We are given $f: X \rightarrow P(X)$, $f(x) = X\backslash\{x\}$, and $X$ is a set. Is the function injective, surjective, bijective?

I am working on this problem in a beginners set theory class. I believe the function is injective but not surjective, thus is it not bijective. We can show it is injective by letting $f(x) = f(x')$. ...
0
votes
1answer
41 views

Field Proofs with Multiplicative Inverses

I've been staring at these two for a while and I can't come up with anything concrete to start. Hints on how to begin would be greatly appreciated, full solutions are not necessary (and preferably ...
0
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0answers
46 views

Proving that $3 = 9^{-1} \pmod{26}$

Prove that $3$ is the multiplicative inverse of $9 \pmod {26}$ $$\quad26\quad1\quad0\\2\quad9\quad0\quad1\\\;\;1\quad8\quad1\quad{-2}\\\quad\;1\quad-1\quad3$$ Hence $3$ is the multiplicative inverse ...
0
votes
3answers
49 views

Game Theory Voting Utilities

! So far, I've managed to come up with this solution: ! But as far as here...I can convert this into payoffs, however I'm unsure of how to figure out the Nash equilibria as when we convert from ...
1
vote
0answers
24 views

Prove that a mixed strategy in two player, zero sum, matrix game must exist (alternative proof)

So I am having a trouble with this game theory proof. I feel pretty good with my answer for part 1, but I am not really sure how to get started on the rest of it. Any help would be appreciated. Let ...
0
votes
4answers
49 views

Prove that for all integers a and b, a + b and a − b are either both odd or both even.

Stumped on this proof. I've only been able to figure it out assuming that both a and b are even: $a = 2k$ and $b = 2n$ $2k + 2n = 2(k + n)$, definitely even. $2k - 2n = 2(k - n)$, also definitely ...
1
vote
2answers
50 views

Proving by induction, if the base case fails to meet the main condition, what do we do?

I have to determine the number $x$ of subsets with odd cardinalities of a set $S$ and then prove that I'm correct. I determined the number $x$ is obtained using the formula $2^{n-1}$ where $|S| = n$. ...
0
votes
3answers
29 views

Bounded Sequences and Extrapolation of Convergence From Related Sequences

I'm considering some sequence $S_n$ which is bounded, and I want to prove that $S_n/n$ is convergent. I'm thinking that I could simply take $lim_{n \to \infty} S_n/n$ and simplify this to $(lim_{n \to ...
0
votes
0answers
34 views

Proving that a value is a multiple of 11

I need to prove that 12**n - 1 is a multiple of 11 for every value of n (part of N). This is clearly a proof-by-induction problem. My base case is 0, where I assume n = 0 will give a resulting 0 ...
2
votes
2answers
53 views

Showing that $(\mathbb{R} \setminus \{ 0 \}, \, \times) \not \cong (\mathbb{C} \setminus \{ 0 \}, \, \times)$

I'm trying to show that $(\mathbb{R} \setminus \{ 0 \}, \, \times) \not \cong (\mathbb{C} \setminus \{ 0 \}, \, \times)$ as follows: note that there exists an element (namely $i$) in $\mathbb{C} ...
1
vote
0answers
28 views

Prove the congruence $pB_{p-1} \equiv -1 \pmod p$ for Bernoulli numbers. [duplicate]

I need to prove that: If $p$ is prime greater than or equal to five, then $pB_{p-1}$ belongs to the p-integers and more over: $$pB_{p-1} \equiv -1 \pmod p$$ Hint:Put $N=p$ in the Faulhaber´s ...
1
vote
0answers
29 views

How to prove this

Prove that if $A^x+B^y=C^z$ where $A,B,C,x,y,z$ are positive integers and $x,y,z$ are all greater then $2$ then $A,B,C$ must have a common prime factor. I heard about this problem a long time ago. I ...