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.

learn more… | top users | synonyms

0
votes
1answer
59 views

Using contrapositive to Prove that if an average of a thousand numbers is less than 7, then at least one of the numbers being averaged is less than 7 [duplicate]

so I know that the contrapositive will be something like; If all the numbers are greater than or equal to 7, then the average cannot be less than 7. How do i go about proving it from there? or is ...
2
votes
2answers
81 views

Prove that if an average of a thousand numbers is less than 7, then at least one of the numbers being averaged is less than 7 [closed]

I tried proving this by contraposition, by saying, "If every number that is being averaged is greater than 7, then the average of a thousand numbers is less than 7." This seems easier to prove, but I ...
0
votes
1answer
46 views

How to formal prove |set 1| is less than or equal to |set 2| + |set 3|

I could prove |set 1| $\le$ |set 2|by defining a function $f$ such that $f$ is a one-to-one function Suppose X = {1, 2, 3} and Y = {D, B, C, A} or X = {1, 2, 3, 4} and Y = {D, B, C, A} and ...
1
vote
3answers
201 views

How can I prove that two vectors in $ℝ^3$ are linearly independent iff their cross product is nonzero?

Here's my attempt: Let $𝒙 = (x_1, x_2, x_3)$ and $π’š = (y_1, y_2, y_3)$ The cross product of $𝒙, π’š$ is $π’™β¨―π’š=(x_2y_3-x_3y_2, x_3y_1 - x_1y_3, x_1y_2 - x_2y_1)$ And linear independence of $𝒙, ...
1
vote
2answers
192 views

Proof by induction and combinations

I think I am stuck on this, I am not sure if I'm going down the correct path or not. I am trying to algebraically manipulate $p(k+1)$ so I can use $p(k)$ but I am unable to do so, so I am not sure if ...
3
votes
2answers
68 views

Columns of $AB$ are independent $\rightarrow$ Columns of B are independent

Let $A\in M_{mxn}(F)$ and $B\in M_{nxk}(F)$ Columns of $AB$ are independent $\rightarrow$ Rank(AB)=k on the other hand $Rank(AB)\leq min(Rank(A),Rank(B))$ therefore $k \leq min(Rank(A),Rank(B))$ due ...
0
votes
1answer
23 views

Example of a set with some caracteristics.

My question this time is, I am asked to find an example of a subset of $\mathbb{R}^{n}$ such that it is closed but not bounded, and all continous function defined there are also uniform continous, but ...
1
vote
2answers
39 views

Find all perfect squares of the form aaa…a (n digits) bbb…b (n digits)

Find all perfect squares of the form $\underbrace{a \ldots a}_n \underbrace{b\ldots b}_n$. This is my homework for Discrete Mathematics. All I'm asking for is to better understand what exactly the ...
0
votes
1answer
22 views

is this possible, and why?

So I am studying a proof, and at one point, it states that for all $x$ and $y$ for whom it holds true to $|x-y| < \pi$ and $y \ge x$, we can pick an x so $$2k\pi \le x < 2k\pi + 2\pi$$ and then ...
5
votes
2answers
109 views

Prove $n^2+4n+3$ is not prime for $n \in \mathbb{Z}^{+}$.

I am trying to write a proof for this theorem: For every positive integer $n$, $n^2+4n+3$ is not a prime. Proof: Let $n \in \mathbb{Z}^{+}$. Note that $$n^2+4n+3=(n+1)(n+3)>1\text{,}$$ and ...
1
vote
1answer
46 views

How to prove uniqueness of |a - c| = |b - c|

So I'm working on a problem, and the problem asks to prove that |a - c| = |b - c| has one unique integer solution for any odd integers a and b. I have proven that there exists a number (a + b)/2 ...
4
votes
1answer
58 views

Lemma for $d(p,q)$ to be a metric for $\mathbb{R}^1$.

I have this exercise where I have to show whether or not $d(p,q)$ is a metric for $X = \mathbb R^1$, and I have come up with this lemma to help me with it. I need help proving it (if it is true). The ...
0
votes
2answers
59 views

Complex Analysis: Show that $Arg(z)$ is discontinuous at each point on the nonpositive real axis.

That's the question. I'm not entirely how to prove, per se, that $Arg(z)$ is discontinuous at every point on the nonpositive real axis. By defintion, $- \pi < Arg(z) \leq \pi$, so I'm not sure ...
-1
votes
1answer
83 views

For each n the belongs to N, there exists m that belongs to N such that m > n

I am learning proofs with $\mathbb N$. Here is my proposition: For each $n \in\mathbb N$ there exists $m \in\mathbb N$ such that m > n. Here are my axioms: a)If $m,n \in\mathbb N$ then $m + n ...
0
votes
1answer
23 views

Algebraic Manipulation in a Proof

I am in the middle of a stats-related proof and got stuck in an algebraic manipulation. The source formula goes $e^{tn}(pe^{-t}+q)^n$ and my target result is to simplify that into $(qe^t+p)^n$. I'm ...
0
votes
1answer
26 views

$f$ is defined and positive in $[0,1]$ for $1>a>0 \implies \inf f((a,1])>0$ if $f$ continuous on the right at $x=0$ proof that $\inf f([0,1])>0$

I have this problem : $f$ is defined and positive in $[0,1]$ for $1>a>0 \implies \inf f((a,1])>0$. if $f$ continuous on the right at $x=0$ proof that $\inf f([0,1])>0$ My proof Since ...
2
votes
2answers
52 views

Verfiying a proof in PDE Theory

I am a student in Computer Science and I have gain an interest in PDE Theory. I got a book on this subject authored by L.C. Evans. In the book problems are solved in a analytic way. I was wondering ...
0
votes
1answer
67 views

Proving $\lim\frac1{x_n} = -\infty$ if $\lim x_n = 0$ [closed]

Suppose $x_n < 0$ for all $n$. If $\lim x_n = 0$, prove that $\lim\frac1{x_n} = -\infty$.
0
votes
3answers
83 views

Show that if $\bar{A} \cap \bar {B} \subseteq C$ then $A \cup B \cup C$ is the universal set

For any subset $A$ of the universe $U$, we denote $\bar{A}$ as $U - A$. Show that for any subsets $A, B, C$ of the universe $U$, if $\bar{A} \cap \bar {B} \subseteq C$ then $A \cup B \cup C = U$. ...
0
votes
2answers
50 views

How to prove that the composition of two surjective functions is surjective [duplicate]

I know that the map $f:A\to B$ is a surjective function (onto) if for all $b$ in $B$, there exists an $a$ in $A$ such that $f(a)=b$ But I am having trouble getting started with this proof since it ...
0
votes
1answer
84 views

I need to prove a proposition about natural numbers

I am learning proofs with $\mathbb N$. Here is my proposition: For each $n \in\mathbb N$ there exists $m \in\mathbb N$ such that m > n. Here are my axioms: a)If $m,n \in\mathbb N$ then $m + n ...
2
votes
1answer
71 views

Integral questions [closed]

three questions about integrals for you. 1)'Given that f(x) is even on $R$, show that $F(x) = \int^x_0f(t)dt $ is odd.' Was I right in doing the following: Given that $f(x)$ exists, this implies ...
1
vote
1answer
98 views

Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$

Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$ Could someone explain/define the multiplication here for me so that I may attempt this problem. Thank you ...
2
votes
3answers
55 views

Let $H$ be a subgroup of a group $G$ and $a,b \in G$. Prove that the following

Let $H$ be a subgroup of a group $G$ and $a,b \in G$. Prove that the following statements are equivalent. (1) $a^{-1}b \in H$ (2) $b^{-1}a \in H$ (3) $aH=bH$ So, I started with (3). $aH=bH \to ...
1
vote
1answer
129 views

Very confused $x^2$ has 1 root

How would you prove that $x^2$ has exactly 1 root using the rolle's theorem? If there's f(a)>0 and f(b)=0 then f(a) does not equal f(b) does that proof that there's one root?
2
votes
4answers
78 views

Proving $\arctan x > \frac x{1+x^2}, \forall x >0$ with a helper function

Prove $\arctan x > \frac x{1+x^2}, \forall x >0$ There's the approach using Lagrange's, but is it also possible to define a function like so? $f(x)=\arctan x - \frac x{1+x^2}$, take the ...
3
votes
1answer
346 views

How to deal with this double summation?

I'm stuck with the proof of this result: $$2^n = \sum_{t=-\frac{n-1}{2}}^{\frac{n-1}{2}} \binom{n+1}{\frac{n+1}{2} + t} \sum_{k=\vert t \vert}^{\frac{n-1}{2}} \binom{\frac{n-1}{2}+k}{k} ...
2
votes
1answer
72 views

Uniform Convergence of $nx^{2}(1-x)^{n}$ on $[0,1]$

Uniform Convergence of $nx^{2}(1-x)^{n}$ on $[0,1]$ My attempt: criterion: suppose $f_n:I\to\ J$ is a sequence of functions which converges point wise to a function $f$, then the convergence is ...
6
votes
2answers
460 views

Proof of whitney's embedding theorem?

While learning about the rigorous definition of manifolds, my text mentions that any $n$-dimensional manifold can be embedded in $\Bbb{R}^{2n}$, which is called Whitney's embedding theorem. I have ...
2
votes
2answers
83 views

Prove $A-I $ is not invertible. [closed]

If A=($a_{ij}$) is a nxn matrix of real numbers such that $\sum_j^n$ $a_{ij}$=1 for each i, show that the matrix A-I is not invertible. I am not very good at proofs and have previously heard you ...
1
vote
6answers
79 views

Prove that the limit of $a_n = \frac{2n^3+n^2}{(n+2)^3}$ is $2$.

if $a_n = \dfrac{2n^3+n^2}{(n+2)^3}$, prove that the limit of $a_n$ (as $n$ tends to infinite) is $2$ using the definition of a limit. My attempt was $\left |\dfrac{(2(n^3))+(n^2)}{(n+2)^3} - ...
0
votes
1answer
45 views

Prove the sequence $\{\frac{1}{a_n}\}\,\, (n\geq 1)$ is Cauchy.

Given that $\{a_n\}$ ($n\geq 1$) is a Cauchy sequence, prove that if there exists an $r$ greater than $0$ such that for all $n$ in the natural numbers $a_n$ is greater than $r$, then ...
1
vote
2answers
37 views

I have a question about “if and only if” propositions.

I have the following proposition: $m - n = p - q$ if and only if $m + q = n + p$ From what I understand, $A = B$ if and only if $C = D$ means two statements: if $A = B$ then $C = D$ and if $C = D$ ...
-1
votes
3answers
69 views

If $a \equiv b$ (mod 2n), then $a^2 \equiv b^2$ (mod 4n)

How would I go about proving: If $a \equiv b$ (mod 2n), then $a^2 \equiv b^2$ (mod 4n)? I already tried proving $a+b = 2nk$ for some integer k, and that was pretty straightforward. But when I try to ...
1
vote
2answers
50 views

A basic proof: $\forall a,b\in\mathbb{Z}$, if $a\left.\right|b$ then $a^2\left.\right|b^2$

I must state whether the following is true or false on my homework (yes, this is a homework problem, so I would appreciate it if you would only give hints or suggestions and not write out the total ...
2
votes
3answers
27 views

Prove that $z_n \rightarrow z_0$ if and only if $\bar{z_n} \rightarrow \bar{z_0}$ as $n \rightarrow \infty$.

In my Complex Analysis course, I'm supposed to prove that. I'm not really too sure where to start, however. Any pointers? Thanks. :-)
-1
votes
3answers
89 views

Prove that $(x_1+x_2)^2 \neq x_1^2+x_2^2$

For $x_1, x_2 \in \mathbb{R}$ there is the rule: $(x_1\times x_2)^2=x_1^2\times x_2^2$. How can I prove, that this rule doesn't apply for: $(x_1+x_2)^2$?
-1
votes
1answer
209 views

The union of two sets A, B is the set AUB. Prove that if A and B are nonempty bounded subsets of R, then AUB is bounded and supAUB = max{supA, supB}.

Proof: If A,B C R and are nonempty and bounded, ==> There exists a least upper bound M s.t: x <= M for all x in A or x in B, by the Completeness Axiom. If A and B are bounded ==> AUB is bounded. ...
0
votes
1answer
67 views

Proof of an alternate form of the triangle inequality

Since it is all positive squaring does not change anything. So: $$ (a_1^2 + \cdots + a_n^2) + 2\sqrt{(a_1^2 + \cdots + a_n^2)(b_1^2 + \cdots b_n^2)} + (b_1^2 + \cdots + b_n^2) \ge (a_1 + b_1)^2 + ...
0
votes
1answer
22 views

Gradient of composition

Consider a function $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ defined by $g(x) := \langle a,x \rangle = a_{1}x_{1}+ ....+ a_{n}x_{n}$, for some $a \in \mathbb{R}^{n}$. Then consider the function ...
1
vote
2answers
48 views

Proof by induction $n^2-2n-1>0$ for $n \ge 3$

I want to use induction to prove that $n^2-2n-1>0$ for $n \ge 3$ Base case: $3^2-2(3)-1>0$ $ \space \checkmark$ Inductive step: $(n+1)^2-2(n+1)-1>0$ $\iff n^2+2n+1-2n-2-1>0$ $\iff ...
0
votes
0answers
68 views

Euler's theorem of homogeneous function (I dont understand the proof)

Here is a link for those who want to take a look at the theorem. (http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf) I considered a function g(t)=f(tx) for fixed x and I took a ...
0
votes
0answers
11 views

Are these correct solutions for two prepositions?

I have two propositions that I need to prove using the addition and multiplication axioms as well as two propositions that I have proven: A) For all $m, n \in\mathbb Z, (-m)(-n) = mn$ B) For all $m ...
2
votes
0answers
23 views

Give counterexamples of some assertions in probability

I have the following exercise: An event $F$ is said to carry positive information about an event $E$, and we write $F \uparrow E$ if $P(E|F) \ge P(E)$ Prove or give counterexamples of the following ...
2
votes
2answers
59 views

Prove $\lim_{x\to \infty} \frac{4x^2 - 7}{2x^3 - 5} = 0$ using $\epsilon$-$N$ limit definition

I am having difficulties manipulating the problem so that I can find a $N$ value to choose. Suppose $x > N$, then $$\left|\frac{4x^2 - 7}{2x^3 - 5}\right| \leq \frac{4x^2}{|2x^3 - 5|} + ...
0
votes
0answers
33 views

Proving the additive inverse of a sum is the sum of the additive inverses

I have this proposition to prove: For all $m, n, \in\mathbb Z$: $-(m + n) = (-m) + (-n)$ Proof: \begin{align*} -(m + n) + (m + n) &= 0 + 0\\ -(m + n) + (m + n) &= (-m) + m + (-n) + n\\ (-m) + ...
0
votes
0answers
48 views

Proof technique question related to Euler's theorem for homogeneous function

I am trying to prove Euler's theorem for homogeneous function. Actually, proof is given http://people.hss.caltech.edu/~kcb/Notes/EulerHomogeneity.pdf What I don't understand is that proof is based ...
0
votes
1answer
40 views

Events that carry negative information

I want to answer the following exercise: An event $F$ is said to carry negative information about an event $E$, and we write $F \downarrow E$ if $P(E|F) \le P(E)$ Prove or give counterexamples of ...
1
vote
2answers
35 views

I am stuck at this proposition

I am stuck at this proposition: Let $x \in\mathbb Z$. If $x \cdot x = x$, then $ x = 0$ or $x = 1$. I am learning the "or" statement and would greatly appreciate any hint. Thank you! Here are the ...
1
vote
1answer
51 views

injective functions require functions whose composition equal the identity function

I'm trying to prove: Let $A, B$ be sets and let $f : A\to B $ be a function. Prove that if $f$ is one to one then there is a function $g : B\to A$ so that $g(f)=id_A$ Here is what I have, Suppose $f ...