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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0
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1answer
16 views

Prove that Autonomous are invariant under time translation

Reading my way through a big boy ODE book, and the authors write It is clear that if $\varphi(t)$ is a solution to $x'=f(x) \quad x(t_0)=x_0$, then clearly $\varphi(t+t_0)$ is a solution to $x'=f(x) ...
1
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2answers
23 views

Prove the existence of a greatest lower bound of $X$ if $X \subset \mathbb{R}$ is a non-empty set that is bounded below

Attempt: Let $C \subset \mathbb{R}$ be the set of all lower bounds of $X$. Since $C$ is not empty and bounded above, every $x \in X$ is an upper bound of every element $c \in C$. Thus, there exists ...
2
votes
1answer
29 views

Upper and/or lower Bound for Numbers of different topologies on the set $\{1,…n \}$

As the title says I am looking for upper and lower bound for the cardinality of different topologies on a set $\{1,....n\}$ for natural n! Are there some known bounds? My teacher says that there no ...
0
votes
1answer
30 views

How to Prove: If $A$ and $B$ are subfields of a field $F$, then $\{b+a|b\in B, a\in A\}$ is also a subfield of $F$.

I haven't been able to find any counterexamples for either of the two. (1) seemed intuitively true but I had my doubts on (2) and couldn't find one. If there aren't any counterexamples, how can I go ...
1
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0answers
34 views

Proof Strategy: For all nonzero complex numbers $z$ and all nonzero rational numbers $a$ and $b, \mathbb Q (az+b)=\mathbb Q(z)$

I am having trouble proving (or finding a counterexample but I believe it to be true) the following. Prior to this I did some problems such as: Show that $\mathbb Q (-3+i\sqrt{2},2-\sqrt{8})=\mathbb Q ...
2
votes
2answers
22 views

If $S \in L(X,Y)$ and lim$_{r \to 0}\frac{\|Sr\|}{\|r\|}=0$,then $S=0$.

Here is a lemma whose proof is as under: If $S \in L(X,Y)$ and lim$_{r \to 0}\frac{\|Sr\|}{\|r\|}=0$,then $S=0$. Proof: The condition lim$_{r \to 0}\Big(\frac{\|Sr\|}{\|r\|}\Big)=0$means that ...
1
vote
2answers
37 views

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected Suppose $G$ is disconnected then $\exists A,B$ relatively open disjoint sets so that $A\neq ...
0
votes
3answers
42 views

If $f$ is 1-1, prove that $f(A\setminus B) = f(A)\setminus f(B)$

I'm having a tough time with this one. Here's the background: Let $X$ and $Y$ be sets, let $f:X\rightarrow Y$ and let $A,B\subseteq X$. For this proof, we also assume that $f$ is 1-1. I've already ...
1
vote
3answers
63 views

Proof that $\nabla(ab) = a\nabla b + b\nabla a$

I am trying to prove the following are equivalent: $a(x,y,z)$ and $b(x,y,z)$ $$\nabla(ab) = a\nabla b + b\nabla a$$ So looking at the left side: $\nabla(ab)= \cfrac{\partial ab}{\partial x} + ...
-1
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0answers
32 views

$a$, $b$, $c$ > 0, if $\frac ab + \frac bc + \frac ca$ is an integer, prove that $abc$ is a cube of an integer [duplicate]

$a$, $b$, $c$ are not equal to $0$, if $\frac ab$ + $\frac bc$ + $\frac ca$ is an integer, prove that $abc$ is a cube of an integer. I really have no idea, appreciate any kind of help
0
votes
1answer
26 views

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$. My attempt: Let $x_n\to -\infty$ and $\epsilon\gt 0$. By the Archimedean Principle pick $N\in \mathbb N$ such that ...
1
vote
3answers
27 views

Show that for all real numbers a and b, ab <= (1/2)(a^2+b^2)

so as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b$ $\epsilon$ $R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ...
0
votes
3answers
53 views

Find all $n\in \mathbb N$ such that $\sqrt{n+7}+\sqrt{n}$ is rational.

Find all $n\in \mathbb N$ such that $\sqrt{n+7}+\sqrt{n}$ is rational. By inspection it is pretty easy to see that the only $n$ that will work is $n=9$. Because the distance between perfect squares ...
2
votes
1answer
65 views

How to prove a function is not onto?

Let $f : Z\to Z$ be the function defined by $f(x) = 3x + 1$. Prove that $f $ is not onto, using a proof by contradiction. (Choose an integer $n$, and then prove ($\forall m \in Z$)($f(m) ≠ n$) by ...
2
votes
0answers
14 views

Given the sets $X$ and $Y$ in the real numbers with least upper bounds $a$ and $b$ respectively, prove that $a+b$ is the least upper bound for $X + Y$

I've seen this proof done other ways and I wonder if my way is right. It's very similar to the $\epsilon > 0$ approach I've seen elsewhere but uses a contradiction: Let $X$ and $Y$ be sets of real ...
0
votes
2answers
26 views

The boundary of the union of two sets is a subset of the union of boundaries

I'm stuck on trying to get this proof started. I want to prove that $\delta(S_1 \cup S_2)\subset \delta S_1\cup\delta S_2$, where $S$ is some set. I don't need a full proof, just a hint to get ...
1
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2answers
23 views

induction for idempotent matrix : $P^n = P$

Given that $P^2 = P$ how do i prove by induction that $P^n = P$? I have tried the following: we know that $P^k = P$ holds for $k = \{1,2\}$. If we now take $k=3$: $$ \begin{align} P^3 &= ...
1
vote
3answers
175 views

prove this inequality with log and positive value “x”

How do I prove that for every positive $x$ , $1-x \le -\log{x}$ Can I use convexity somehow?
0
votes
1answer
82 views

Prove ${2n\choose n}=\sum\limits_{k=0}^n {n\choose k}^2$ [duplicate]

Prove ${2n\choose n}=\sum\limits_{k=0}^n {n\choose k}^2$ My Approach: I will be making use of $$\tag 1\quad{m+n\choose r} = {m\choose 0}{n \choose r} + {m\choose 1}{n\choose r- 1} + ...
0
votes
1answer
36 views

Practice Examples of Proofs by Induction, Direct/Indirect Method

I'm learning about proofs in school, quite a few different sorts (but not geometry ones), but the teacher is teaching by slides mainly, not books. The main ones are proof by ...
0
votes
0answers
41 views

Induction and Maximum Principle

I wish to show that the following two assertions are equivalent: (Principle of Mathematical Induction) Let $S$ be a nonempty subset of the set of non-negative integers satisfying the following two ...
1
vote
1answer
45 views

For any two Ideals $A$ and $B$,$A+B=\langle A \cup B \rangle$

Below is the proof of : Prove that for any two ideals $A$ and $B$ of ring $R$,$A+B=\langle A \cup B~\rangle$ . Proof: By theorem (for any two ideals of a ring $R$ ,then the set $A+B$ is an ...
0
votes
0answers
54 views

When to use weak, strong, or structural induction?

For weak induction, we are wanting to show that a discrete parameter n holds for some property P such that P(n) implies P(n+1). For strong induction, we are wanting to show that a discrete parameter ...
1
vote
1answer
57 views

Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$

Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$ My Approach Let $x_k$ be one element in a set of $n$ elements. $n-1\choose r-1$ $=$ the number of unique groups of $r$ containing ...
0
votes
1answer
25 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
vote
1answer
20 views
0
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0answers
16 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.
1
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0answers
14 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
votes
3answers
109 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
0answers
10 views

show that for the system to be consistent we must have b2 = b3 - 2b1 [closed]

The first problem, #4. I dont know how to approach this problem, I can see how the statement b2 = b3 - 2b1 is true but how do I prove it must be true?
0
votes
1answer
12 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
0
votes
2answers
29 views

Proving antisymmetry within matrices [closed]

If $A$ is a $3\times 3$ antisymmetric matrix of real numbers, how can I prove that $A^2$ is a symmetric matrix?
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2answers
33 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.
4
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2answers
60 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
42 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 ...
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votes
0answers
44 views

proving an inequality [closed]

If $0 < c < 1$, prove that there is a positive number $h$ such that $\displaystyle 0 < c^n < \frac{1} { 1+nh}$ . With the same number $h$, formulate a non-trivial inequality for $(nc)^n$.
0
votes
3answers
66 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
75 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
68 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
vote
0answers
81 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
vote
1answer
25 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
votes
4answers
291 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
69 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. ...
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votes
0answers
68 views

Proving Cartesian sets are subsets to one another

Prove the following implication using proper proof techniques: If S ⊆ T, and T ⊆ V, then S x S ⊆ T x V I know that all x's belongs to S, implies x belongs to T and therefore, to V. Also, I know the ...
-1
votes
1answer
32 views

Transitivity of subsets [closed]

Let S, T, and V be any 3 sets. Prove the following implication, using full proof techniques: If S ⊆ T and T ⊆ V then S ⊆ V .
10
votes
1answer
113 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
153 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
vote
1answer
20 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
47 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.