For questions about the formulation of a proof, not about the mathematics behind it.

learn more… | top users | synonyms

5
votes
3answers
176 views

How to integrate $\int_{-3}^3 (x^2-3)^{3} \,dx$ without expanding the polynomial?

How can I integrate: $$\int_{-3}^3 (x^2-3)^{3} \,dx,$$ neither expanding the polynomial nor using the relationship between integral and derivatives? I mean, there is a way to compute this integral ...
3
votes
4answers
87 views

Proof writing: $\sum_{n=1}^{\infty}| a_n|<\infty $ implies $\sum_{n=1}^{\infty} a_n^2<\infty $.

Let $\sum_{n=1}^{\infty} a_n $ be an absolutely converging series. By definition, this means $\sum_{n=1}^{\infty} \lvert a_n\rvert $ converges. We want to show that $\sum_{n=1}^{\infty} a^2_n $ ...
1
vote
0answers
52 views

Show $\mathbb{N}^{\{0,1\}}$ is uncountable with a hint

Let $\mathbb{N}^{\{0,1\}} :=\{f: \mathbb{N} \to \{0,1\}\}$ is uncountable I have never heard of the table approach, and all the proofs say uncountability of $\mathcal {P}(\mathbb{N})$ I have seen so ...
1
vote
4answers
42 views

Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric.

How can i proof the following statement: "Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric." i tried to work out the properties of a matrix to be ...
2
votes
0answers
64 views

Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold.

Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold. The boundary is defined by $z = \sqrt{x^2+y^2}$. I'm trying to think how to approach ...
0
votes
1answer
37 views

Prove $\forall n \in\mathbb{N}, (n|105 \wedge n|70) \implies 5|n$

I know the definition of divides into is $$a|b \equiv \exists a\in\mathbb{Z}, b = ac$$ however I'm not sure how to manipulate this to prove $$\forall n \in\mathbb{N}, (n|105 \wedge n|70) \implies 5|n$$...
1
vote
1answer
53 views

Proving $n < 2^n$ by Cantor's theorem

So we know Cantor's Theorem is of course. For any set $S$, the power set $P(S)$ has a strictly greater cardinality, $\iff \#S < \#P(S)$. We seek to prove $n < 2^n$ using this information. I ...
0
votes
3answers
33 views

Proving the u-substitution formula

Let $g: [a, b] \rightarrow [c,d] $ be continuously differentiable and $f: [c,d] \rightarrow \mathbb{R} $ continuous. Prove that 􏰀$\int_{a}^{b} f(g(x))g'(x) dx $ = $\int_{g(a)}^{g(b)} f(t)dt $ ...
4
votes
3answers
63 views

Disprove: $f\circ g = f \circ h \implies g=h$ for a surjective function $f$

I tried using a very specific counterexample here where I select a surjective function for which the compositions are equal but the functions within are not. This is probably off-base, but it's what ...
0
votes
1answer
17 views

Show two notions of dense are equivalent

This question follows from another one Topology proof: dense sets and no trivial intersection Show that given a topological space $(X, \mathcal{T}), D \subseteq X$ Then $D$ is dense iff $\...
4
votes
2answers
145 views

Proof of a statement about eigenvalues and eigenvectors.

How can i proof the following: Let $\mathbb L: V\rightarrow V $ be a linear mapping. Let $v_1,v_2,..,v_n$ non-zero eigenvectors with eigenvalues $c_1,c_2,..,c_n$ respectively, also let the ...
0
votes
1answer
17 views

How to proof that the set of all $X$ such that $X.A{\ge} c$ to some real number c is convex?

How can i proof the following statement: " Let $\mathrm A\in \mathbb R^{n}$ and $\mathrm c\in \mathbb R$, the set $\mathbb S$ of all elements belonging to $\mathbb R^{n}$ and satisfying the ...
3
votes
1answer
23 views

For $f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(x)=ax+b$, for what $a,b \in \mathbb{R}$ is $f$ a bijection?

I asked a similar question here. This question has different parameters however as you can see. For $f: \mathbb{N} \rightarrow \mathbb{N}$ given by $f(x)=ax+b$, for what $a,b \in \mathbb{R}$ is $f$ a ...
0
votes
1answer
35 views

If $f$ is injective, then $f(X\backslash A) = f(X) \backslash f(A)$

Given $f:X \to Y$ injective, $A \subseteq X$, then $f(X\backslash A) = f(X) \backslash f(A)$ I have spent a long time looking at this problem but I have not found a good way to approach this. Here ...
1
vote
2answers
30 views

Prove $f: A \rightarrow B$ is strictly injective, $\implies$ $f^{-1}$ is a function and dom $ f^{-1} \subset B$

The question I have about this proof is that, do I need to choose a specific function $f:A\rightarrow B$ that is not injective but surjective? Will I lose generality if I do? For instance, I was ...
1
vote
0answers
29 views

Show the Heaviside step function is continuous in $(\mathbb{R}, \mathcal{T}_\text{lower limit})$

Given $(\mathbb{R}, \mathcal{T}_\text{lower limit})$ where lower limit topology $\mathcal{T}_\text{lower limit} = \mathcal{T_\mathcal{B}}$ where $\mathcal{B} = \{[a,b) \subseteq \mathbb{R}, a < ...
2
votes
2answers
50 views

Prove $\forall n \geq 10, 2^n > n^3$

Prove $\forall n \geq 10, 2^n > n^3$ base case: $n = 10$ $2^{10} = 1024$ $10^3 = 1000$ $1024 > 1024$. So $P(k)$ holds for $k = n$. We seek to show $P(k+1)$ holds: We know $2^k > k^3$. ...
1
vote
1answer
36 views

Riemann Integrability defined by sequence of partitions

Prove that a bounded function $f$ is integrable on $[a, b]$ if an only if there exists a sequence of partitions $ \left(P_{n}\right)^{\infty}_{n=1} $ satisfying $$ \lim_{n\to\infty} [U(f, P_n) - L(f,...
0
votes
1answer
84 views

Obscure proof that $+$ and $\times$ are continuous?

I am looking for proof of $+$ and $\times$ are continuous operations without using the standard definition of continuity (1. $\epsilon-\delta$, or 2. preimage of open sets or 3. sequential ...
0
votes
0answers
29 views

How to show that $f(X \backslash f^{-1}(Y \backslash C)) \subseteq C$

Let $f: X \to Y$ be a continuous function, and that $C \subset Y$, then claim: $f(X \backslash f^{-1}(Y \backslash C)) \subseteq C$ Attempt: Immediately we run into a problem following: ...
2
votes
0answers
28 views

Powers of two in an infinite sequence of arbitrary integers

I'm not sure how to ask this properly and I haven't seen this problem anywhere yet, but I'm still interested if this can be (dis)proven or not. Consider a finite sequence of the numbers 1, 2, 4, and ...
4
votes
2answers
37 views

Prove $(A \cup B)' = A' \cap B'$

I would like some assistance in verifying this proof? (I understand the last conjecture about "symmetry" is probably shaky, but I just want to know if the first part is right since going backwards ...
7
votes
4answers
117 views

How to show that $f(x) = x^2$ is continuous using topological definition?

I am trying to show that simple continuous functions satisfy topological definition of continuity Recall given $(X, \mathcal{T}), (Y, \mathcal{J}), f$ is continuous if $f^{-1}(V) \in \mathcal{T}, \...
0
votes
1answer
26 views

Equivalence relations proof example?

Let $A$ = {$a,b,c$}. Give an example of a relation on $A$ that is anti-symmetric, reflexive on $A$ and symmetric. The first thing that one must do to proceed with this question is to first define ...
0
votes
1answer
28 views

Show $(\mathbb{R}, \tau_{co-countable})$ is not Hausdorff but every sequence converge to at most one point

Given $(\mathbb{R}, \tau_{co-countable})$, show that it is not Hausdorff but every sequence converges to at most one point. 1. If $(\mathbb{R}, \tau_{co-countable})$ is not Hausdorff, then $\...
1
vote
2answers
50 views

Is the intersection of a relation that is antisymmetric and a relation that is not antisymmetric, antisymmetric

Given a binary relation R,S on set A, assume that R is anti-symmetric. Show R intersection S is anti-symmetric. I started this proof by stating the definition of anti-symmetric with R which is $$ ∀...
1
vote
1answer
31 views

Equivalence relation and class, Proof.

The relation $P$ on $ℝ$ is defined by $xPy$ iff $x^2=y^2.$ (a) Prove that the the relation $P$ is an equivalence relation. (b) Describe the equivalence class of $3$ and $0$. In order to ...
0
votes
3answers
22 views

Nullity and an Isomorphism

I'm working on some introductory proofs in linear algebra, and I think that I could use some help on this particular problem. I want to prove that a linear surjective map $T: R \rightarrow W$ is an ...
9
votes
5answers
147 views

Prove that $2^n$ does not divide $n!$

I want to prove that $2^n$ does not divide $n!$. I was trying by induction and I'm confused about if what I'm doing is right. First I test it with $n=1$. In fact: $$2^1 \nmid 1!$$ So if i take the ...
2
votes
2answers
98 views

Show $\frac{1}{n}$ converges to $0$ using topological definition

I need to use the following definition to show that: $\frac{1}{n}$ converge to $0$ in $(\mathbb{R}, \mathcal{T}_{usual})$ and $(\mathbb{R}, \mathcal{T}_{lowerlimit})$ The defnition is: Given $(...
0
votes
1answer
24 views

Any additional constraint will increase the value of the maximin

In the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $325$, there is the comment "any additional constraint will increase the value of the maximin",...
0
votes
1answer
62 views

Prove that $\int_{a}^{b} f > 0$

I am asked to prove that $\int_{a}^{b} f > 0$. we are given that $f$ is continuous on $[a,b]$ $(\forall x \in [a,b]) \; f(x) \geq 0\; $ and $(\exists x_{0}) \in [a,b] \;s.t.\; f(x_{0}) >0$ my ...
0
votes
0answers
28 views

Proof Related to the Span in linear algebra

I'm working through a proof in my linear algebra textbook, and I think I am a little stuck. I am trying to prove that if $S$ is a non-empty set of vectors in a vector space $V$, the the set $W_s$ of ...
1
vote
0answers
46 views

Is there such a thing as “finite” induction?

I am not sure of the terminology that I am looking for, but I would like to use an inductive proof on the following type of structure. I have something of the form, for every $n \geq 2$ and for any $1 ...
1
vote
2answers
19 views

Is this proof correctly written? Show that the sum of two uniformly continuous functions on $A$ is uniformly continuous on $A$

If $f$ and $g$ are uniformly continuous functions in $A$ show that $f+g$ is uniformly continuous in $A$. Proof: because $f$ and $g$ are uniformly continuous on $A$ we can write $$\forall\varepsilon&...
1
vote
3answers
103 views

How can I show that $\mathcal{B} = \{(a,b)\subset \mathbb{R}\mid a,b \in \mathbb{Q}\}$ is a countable set?

I know that $\mathcal{B} = \{(a,b)\subset \mathbb{R}\mid a,b \in \mathbb{Q}\}$ is a basis on $\mathbb{R}$. I need to show that $\mathcal{B}$ is countable. How can this be done? Attempt: Take ...
0
votes
1answer
28 views

Assume f and g are defined on all of $\mathbb{R}$ and that $\lim_{x\to p} f(x) = q $ and $\lim_{x\to q} g(x) = r $.

(a) Give an example to show that it may not be true that $\lim_{x\to p} g(f(x)) = r$ If we are to assume that f and g are defined on all of $\mathbb{R}$, wouldn't that mean that f and g are ...
2
votes
4answers
113 views

Prove $\ \sin(x) < x \ \ \ \forall x \in(0, 2\pi)$

Problem : Prove $\sin(x) < x \ \ \ \forall x \in(0, 2\pi)$ Now I have a possible solution for this, using limits and the first derivatives of $\sin(x)$ and $x$, but I don't feel it's a very ...
1
vote
1answer
49 views

If a set K $\subseteq \mathbb{R} $ is closed and bounded, it is compact.

If a sequence is closed and bounded, that means it has a sequence that converges. According to the Bolzano-Weierstrauss Theorem (which I am taking for granted), a converging sequence has subsequences ...
1
vote
1answer
28 views

Show that if $K$ is compact and nonempty, then $\sup K$ and $\inf K$ both exist and are elements of $K$.

Show that if $K$ is compact and nonempty, then $\sup K$ and $\inf K$ both exist and are elements of $K$. If $K$ is compact, then by definition it is closed and bounded, and every sequence in $K$ has ...
4
votes
2answers
340 views

Dividing a Checkerboard into L-Shaped Regions

In preparation for the GRE Math-Subject test, and honestly for the fun of it, I've been working through a select number of my texts. The first of which is Saracino's Abstract Algebra text. I was ...
1
vote
1answer
37 views

Prove that F $ \in \mathbb{R} $ is closed if and only if every Cauchy sequences contained in F has a limit that is also an element of F.

I'm a novice at proofs so I like to write out everything, so please bear with me!. I understand that this is a biconditional statement, and I will have to prove it in the forward and reverse direction....
1
vote
2answers
76 views

Why we not check conditions while solving questions?

Note:Down ward problem is just an example to express my question(I know the both solution of problem are insufficient but the first solution is in my 10+2 book and second one is mine which is ...
2
votes
1answer
41 views

If $2^n-1$ is prime, then n is prime - proof involving the Mersenne primes by counterexample

Let $2^n-1$ be prime. Suppose that $n=p_1p_2\cdots p_s$ is composite. Then we have $2^{p_1p_2\cdots p_s}-1$; call it $k$. If $k$ is prime, then its only divisors are $k$ and $1$. But consider the case ...
0
votes
1answer
34 views

Show that $\overline A \cap \overline B \not\subseteq \overline {A \cap B}$ using definition

It is well known that $\overline {A \cap B} \neq \overline A \cap \overline B$ I wish to show that $\overline A \cap \overline B \not\subseteq \overline {A \cap B}$ by using the definition (...
3
votes
6answers
55 views

Using induction to prove for $n ≥ 1, $ $1 \times 5+2\times6+3\times7 +\cdots +n(n + 4) = \frac 16n(n+1)(2n+13).$

This is a very interesting problem that I came across in an old textbook of mine. So I know its got something to do with mathematical induction, which yields the shortest, simplest proofs, but other ...
1
vote
1answer
58 views

How to integrate $\frac{dx}{(x^2+k^2)^m}$, with $m$ positive integer.

How can I integrate: $$\int \frac{1}{(t^2+k^2)^m}\, dt$$ without trygonometric substituition? where $t= (x+(p/2))$ and $k= (1-(p^2/4))$ coming from an equation with complex roots: $x^2 + px + q.$ ...
3
votes
1answer
49 views

For every integer $m\geq 0$ let $I_m=\int_0^1x^m\left(x^2 -1 \right)^5dx$. Prove that for $m\geq 2$ $I_m= \frac{m-1}{m+11}\,I_{m-2}.$

This is a very interesting problem that I came across in an old textbook of mine. So I know its got something to do with integrals, but other than that, the textbook gave no hints really and I'm ...
1
vote
3answers
53 views

How to show the usual topology is finer than co-finite topology on $\mathbb{R}$

I have solved a bunch of problems where the basis is used to quickly deduce which topology is finer than which. However, I do not know the basis of co-finite topology. What is the straight ...
0
votes
1answer
30 views

A set of discontinuities?

$ \mathbb{Q} $ is countable, so we can list its elements. Let $ \mathbb{Q} = \{r_1 , r_2, ...\}$ Define $f: \mathbb{R} \to \mathbb{R} $ by the following rule: $ f(x) = \begin{cases} 0 & x \...