For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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2
votes
1answer
30 views

Show that if $A$ is any subset of a topological space $(X,T)$, then $Int(A) = \complement ( \overline {\complement (A)})$

Show that if $A$ is any subset of a topological space $(X,T)$, then $Int(A) = \complement ( \overline {\complement (A)})$ My reasoning went as follows: $\overline {\complement (A)} = \complement (A) ...
0
votes
1answer
36 views

Prove that in $\Bbb R$, $Int ([0,1]) = (0,1) $

Basically I need to show $Int([0,1]) = (0,1)$ meaning that I need to show that: $(0,1) = \bigcup_{a \in A}a$ Where for all $a \in A, a = (b,c)$ where $b,c$ real numbers such that $0 <b <c ...
1
vote
2answers
68 views

Proving the geometric series $\sum_{i=0}^n r^i = \frac{1-r^{n+1}}{1-r}$ by induction for $n\geq 1$

Let $P(n)$ be the statement $$ P(n) : \sum\limits^{n}_{i=0}r^i = \dfrac{1-r^{1+n}}{1-r}\text{ for all }n \in \mathbb{N}\text{.} $$ I am stuck at the base case: $$P(1):1 + r = ...
2
votes
2answers
48 views

Proof : If $\{a_n\}$ is converge then $\{|a_n|\}$ converge

Proof : If $\{a_n\}$ is converge then $\{|a_n|\}$ converge. My proof : We know that $\{a_n\}$ converge therefore : $$\lim_{n \to \infty} a_n = L$$ All $\epsilon>0$ exist $N \in \mathbb{N}$ so ...
2
votes
1answer
55 views

differentiable on $\Bbb R^{n}× \Bbb R^{n}$

Let $f : \Bbb R^{n} × \Bbb R^{n} → \Bbb R$ be defined by $f(x, y) = x·y$ , Show that $f$ is differentiable on $\Bbb R^{n}× \Bbb R^{n}$ and that $Df(a, b)(x, y) =b · x + a · y$ Here . denotes the dot ...
0
votes
0answers
12 views

How can i explain the symmetry of the function in a more linguistic manner

To understand the convolution of these functions, please read the following wikipedia page I have the following expression $a = <f'(x),f''(x),\cdots>$ and $b = <g'(x),g''(x),\cdots >$ ...
0
votes
1answer
27 views

Proving $f(f^{-1}(D)) \subset D$

Suppose that $f:A \rightarrow B$ and let $D \subset B$. For proving $f(f^{-1}(D)) \subset D$: Let $x \in f(f^{-1}(D))$. Now $f(f^{-1}(D)) \in B$, so $x \in B$. Then $\exists y \in A$ such that $f(y) ...
1
vote
0answers
31 views

Probability and expectation

We are told that Alice and Bob each have a box that contains $n$ balls numbered $1,2,\ldots,n$. They pick balls from their boxes in a series of rounds as follows: In every round, Alice draws a random ...
0
votes
1answer
24 views

Finite Sets Proof on Domains [duplicate]

Just wanted some help with this little proof.: Let X and Y be Finite Sets. Prove that |X^Y| = |X|^|Y|
0
votes
1answer
43 views

Simple Linear Algebra Proof - Determinants

Prove or disprove the following statement: If R is the RREF of A, then det A = det R. So far, I think that this is true, considering A and R are row equivalent, and that the determinant changes as ...
0
votes
0answers
14 views

Closure of a set with specified distance condition

Salam. I've presented the question and my thoughts on it. The question states: Let $S$ be a subset of $\Bbb R$ and $a \in \Bbb R$. Prove that $a \in \overline{S}$ if and only if for each positive ...
1
vote
1answer
42 views

How to proof: The set A × B is infinite if either A or B is infinite and the other set is not the empty set.

I'm having some difficulty trying to prove that. Suppse $A$ is an infinite set and $B$ is the non empty set, I thoght of showing that the cartesian product of both sets have the same cardinality as ...
3
votes
3answers
81 views

Is there a set of $4$ vectors in $\mathbb{R}^3$, any $3$ of which form a linearly independent set?

I have an exercise in my last assignment for linear algebra: Is there a set of $4$ vectors in $\mathbb{R}^3$, any $3$ of which form a linearly independent set? Prove. My answer intuitively is ...
2
votes
3answers
183 views

Proof of the Product Limit Law

Theorem: $$\lim_{x \to a} f(x) = L$$ $$\lim_{x \to a} g(x) = M$$ Then: $$\lim_{x \to a} f(x) g(x) = LM$$ Obviously, $$|f(x) - L| < \epsilon$$ $$|g(x) - M| < \epsilon$$ But multiplying ...
1
vote
1answer
31 views

Topology generated by the circles on the plane with their centers on a line

Question: Let $S$ be the collection of all circles on the plane which have their centers on the X-axis. If $S$ is a subbasis for a topology $T$ on $\Bbb R^2$, describe the open sets in $(\Bbb R^2, ...
3
votes
1answer
40 views

Prove that $3\log n$ is $O(\exp(0.001n))$

First time posting here. Hi math stack-exchange community! I have a bonus question on my assignment and I am having trouble proving it. The main reason is that I am only limited to using the rules ...
0
votes
2answers
43 views

mathematical induction to establish inequality

Studying for a test in discrete mathematics and I cannot seem to grasp the explanations in the textbook regarding these questions. Using mathematical induction, prove that $$2^n > n^2, \text{for ...
3
votes
0answers
36 views

Much more topology…

These are very similar to my last two questions. I provide them with my thoughts so far: $(1)$ Let $S$ be the collection of all straight lines in the plane which are parallel to the x-axis. If $S$ is ...
-2
votes
1answer
61 views

When $S$, the set of straight lines in $\mathbb{R}^2$, is a subbasis for a topology, what is the topology?

Salam. What are your thoughts on this? Again, please provide as much detail in your explanation as possible... Let $S$ be the collection of all straight lines in the plane $\Bbb R^2$. If $S$ is a ...
2
votes
3answers
55 views

Show $P(1), P(2),…,P(99)$ true statements but $P(100)$ is false.

Provide a sequence of statements, $P(n),$ for $n\in \mathbb{N}$ such that $P(1), P(2),...,P(99)$ are all true but $P(100)$ is a false statement. My try: Let $n\in \mathbb{N}$ and let $0\notin ...
3
votes
1answer
79 views

Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.

Firstly, this is a homework problem so please do not just give an answer away. Hints and suggestions are really all I'm looking for. I must prove the following using mathematical induction: For ...
2
votes
3answers
37 views

Proving that $2n^2 + n + 1 = O(n^2)$ and big O proofs in general

Alright so here's the thing, I'm in a class in Computer Science called Algorithm Analysis and it is required for me to learn Big O, Big Omega, etc. While I sort of understand what this is for, I still ...
4
votes
2answers
45 views

Show that the sum of the $x$-coordinates of three points on the graph of $y = x^2$ whose normal lines intersect at a common point is $0$.

Suppose that three points on the graph of $y = x^2$ have the property that their normal lines intersect at a common point. Show that the sum of their $x$-coordinates is $0$. I've done a bit of work ...
0
votes
1answer
50 views

Prove: $<S>$ $= G$, and every $x \in G$ can be written as $x = s_{1}s_{2}$ with $s_{1}, s_{2} \in S$

I'm trying to solve this problem for my math study, but the things I'm trying don't seem to work. Let $G$ be a finite group, and $S \subset G$ a subset of $G$, with #$S > 1/2 $#$G$ Prove: a) ...
3
votes
1answer
41 views

Prove that $\langle S\rangle$ $= G$, with $S \subset G$ and #$S > 1/p $ #$G$

I'm having trouble with solving this problem: Let $G$ be a finite group of order $> 1$, and $S \subset G$ a subset of $G$, with #$S > 1/p $ #$G$ Where p is the smallest prime factor of the ...
1
vote
1answer
55 views

Topology related question

Salam everyone. If I understand correctly it's site etiquette to typeset math questions in tex? If that is not the case please let me know. Otherwise here is the question : Let $C[0,1]$ be the set of ...
2
votes
1answer
29 views

Another topology question

This is a two part question. The first part, part (i), I present with the solution I reached. The second part, part (ii) is where I need help. (i) Let $B$ be a basis for a topology $T$ on a non-empty ...
0
votes
0answers
27 views

Please help show that the linear transformation of a subspace is equal to itself.

Let $U$ be an orthogonal $n\times n$ matrix, and consider the linear transformation $T : \mathbb{R}^n \to \mathbb{R}^n$ defined by $T(x) = Ux$. Let $W$ be a subspace of $\mathbb{R}^n$ such that $T(W) ...
1
vote
2answers
44 views

Prove that $x_{n+2} := \frac{1}{2}(x_n + x_{n+1})$ converges, if $x_0 = 1$ and $x_1 = 2$?

This question is related to my other question, where I had just to find the limit (which is $\frac{5}{3}$) of the following defined sequence: $$x_0 = 1 \\ \\ x_1 = 2 \\ \\ x_{n + 2} = \frac{1}{2} ...
0
votes
3answers
41 views

Prove that if $x^2+y^2 = z^2$ then $x$ or $y$ is even

I am having trouble proving this. I feel that proof by contradiction would be the best method, although I quickly got stuck after $x=(2k+1), y=(2j+1)$. I expanded so that $4j^2+4k^2+4j+4k+2=z^2$ but I ...
2
votes
1answer
57 views

Bijection on Preordered Sets Implies Homeomorphism

Prove that if $X$ and $Y$ are finite, then the "converse" of one of my other questions Homeomorphism on a Preordered Set is true: if $h: X \to Y$ is bijective and satisfies $\forall a,b \in X, ...
0
votes
3answers
68 views

How to prove that $z_n = 2^n$ converges and therefore has a limit?

I have to prove that the following sequence converges and therefore has a limit: $$z_n = 2^n$$ for $n \in \mathbb{N}$. I have tried to prove it, but I am not seeing exactly what I am doing, that's ...
1
vote
2answers
89 views

Prove that, for $s$ is upper bound of A, $s = \sup A$ iff , if $r < s$, so there exists $x \in A$ such that $r < x \leq s$.

Could someone verify my proof? Definition: Suppose $s \in \mathbb{R}$ and upper bounded $A \subset \mathbb{R}$. For any $x \in A$, we have $x \leq s$. For any $v$ such that $x \leq v$ for any $x$, we ...
2
votes
2answers
17 views

Proving or Disproving statements using sets

I just don't seem to get proofs or set theory so hopefully my question makes sense. I'm not sure when I should or shouldn't use an example to prove or disprove a statement? One example question is, ...
3
votes
2answers
64 views

2014 Putnam A1 Prime number factorial help

Question: Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number. ...
0
votes
1answer
37 views

Prove by contradiction that a circle chord is no longer than its diameter

Can anyone help me with this homework question of mine? I'm actually new to discrete mathematics and to be specific, with proofs. Here's the question, "Prove, by contradiction, that no chord of a ...
0
votes
1answer
39 views

Group Theory - Cyclic Groups & Direct Product

Prove that if $G$ is an infinite group and $H$ is a group then $G \times H$ is cyclic if and only if $G$ is cyclic a $H =$ {${e_H}$}. Solution: I can see that this is true but I don't know how to ...
0
votes
0answers
42 views

How to acquire Mathematical Reasoning & Proof Skills

Dear Math Stack Exchange advisers, I am going to start self-studying the introductory analysis soon by using the textbooks called "Understanding Analysis" by Abbott and "Mathematical Analysis" by ...
1
vote
1answer
28 views

Understanding The Theorem “If there is a trail, then there is a path”

I am given the following theorem and proof: Statement Let $G=(V,E)$ be an undirected graph, $a,b\in V$, $a\neq b$. If there exists a trail(in $G$) from $a$ to $b$, then there is a path (in $G$) from ...
0
votes
0answers
32 views

Show that if $h$ is harmonic , then any mth order partial derivative of $h$ is a linear combination of certain partial derivatives

I want to solve the following exercise Show that if $h$ is harmonic , then any mth order partial derivative of $h$ is a linear combination of $\frac{\partial^{m}h}{\partial z^{m}}$ and ...
4
votes
3answers
84 views

Prove even integer sum using induction

This is a homework problem, so please do not give the answer away. I must prove the following using mathematical induction: $\forall n\in\mathbb{Z^+},\;2+4+6+\cdots+2n=n^2+n.$ This is what I ...
1
vote
2answers
394 views

Proving addition and multiplication

(1)Show that addition and multiplication mod n are associative operations. (2)Show that there are both an additive and a multiplicative identity. (3)Show that multiplication distributes over ...
0
votes
3answers
45 views

Proof the the Arithmetic-Harmonic Mean is expressible as the Geometric Mean

We define the Arithmetic-Harmonic mean of $a,b \in \mathbb{R_+}$ such that \begin{gather*} a_{n+1} = \frac{1}{2}(a_n + b_n) \\ b_{n+1} = \frac{2a_{n}b_{n}}{a_{n} + b_{n}} \end{gather*} Let us also ...
2
votes
1answer
65 views

Countability of a Set

Prove that a set $E$ is countable if and only if there is a surjection from $\mathbb{N}$ onto $E$. Suppose that $E$ is countable. Then there is a bijection from $\mathbb{N}$ to $E$ by definition of ...
0
votes
4answers
45 views

Prove: If $a$, $b$, and $c$ are consecutive integers such that $a< b < c $ then $a^3 + b^3 \neq c^3$.

Prove: If $a$, $b$, and $c$ are consecutive integers such that $a< b < c $ then $a^3 + b^3 \neq c^3$. My Attempt: I start with direct proof. Let $a,b,c$ be consecutive integers and ...
5
votes
5answers
107 views

Prove if $n^2$ is even, then $n^2$ is divisible by 4

I am working on this question Prove for every integer n if $n^2$ is even, then $n^2$ is divisible by 4. prove by contradiction Proof: Since there exists an integer $n$ such that $n^2$ is ...
0
votes
2answers
39 views

Prove if A and B are n x n upper triangular matrices, so is AB

I'm trying to practice proofs for my linear algebra final and I've been stuck on this one for some time. I have $AB = [A\mathbf{b_1} \ A\mathbf{b_2} \ \dots \ A\mathbf{b_n}]$. I can show that ...
4
votes
1answer
43 views

Critique my elementary proof for a set bounded above

Let $A$ and $B$ be two non-empty subsets of $\mathbb{R}$ that are both bounded above. $(i)$Prove that $A ∪ B$ is bounded above and prove $(ii)$ that $\sup(A ∪ B) = \max(\sup(A),\sup(B))$. for ...
-1
votes
2answers
228 views

Proof Involving Rational Numbers

I asked this same question last night and got some answers but still can't make sense of this, normally I'd move on but since I know how to do everything else for the test I'm going to try to get this ...
0
votes
1answer
29 views

Disprove for all integers $a$ and $b$ there exist integers $m$ and $n$ such that $a = m + n$ and $b = m − n$

Use Method of Contradiction to Disprove for all integers $a$ and $b$ there exist integers $m$ and $n$ such that $a = m + n$ and $b = m − n$ I get $\forall m,n \in \mathbb{Z},\exists ...