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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4
votes
2answers
30 views

Help setting out a proof about the circle $x^{2} + y^{2} + 2gx + 2fy + c = 0$

16. Given that the circle $$x^{2} + y^{2} + 2gx + 2fy + c = 0$$ touches the $y$-axis, prove that $f^{2} = c$. So, because the circle touches the $y$-axis, we know that there is a ...
0
votes
1answer
28 views

For every $A\in \mathcal {P}(U)$ there is a unique $B\in \mathcal{P}(U)$ such that for every $C\in \mathcal{P}(U)$, $C\cap A=C-B$

Pls help me out with the proof: For every $A\in \mathcal {P}(U)$ there is a unique $B\in \mathcal {P}(U)$ such that for every $C\in \mathcal {P}(U), C \cap A=C-B$. For the existence part, we have to ...
8
votes
1answer
57 views

Is there a standard name for this “continuous induction” principle?

I am working on a paper, and I want to prove that some statement $P(x)$ holds for every value of a parameter $x \in [0,\infty)$. I plan to proceed as follows: Show that $P(0)$; Show that if $P(x)$ ...
4
votes
3answers
1k views

Derivation of the density function of product of two random variables

I am looking for distribution of product of two random variables. Best I could found so far was this formula from the relevant Wikipedia page: $$ f_Z(z) = \int_{-\infty}^{+\infty} \frac{1}{|x|} ...
2
votes
1answer
65 views

Argue that if a sentence has a proof, then it is a tautology

This is a corollary of the soundness theorem, which states that for a set of formulas $\Phi$ (of propositional logic) and a formula $\alpha$ : $$\Phi\vdash\alpha\Longrightarrow\Phi\vDash\alpha$$ What ...
7
votes
5answers
87 views

Show that if $g \circ f$ is injective, then so is $f$.

The Problem: Let $X, Y, Z$ be sets and $f: X \to Y, g:Y \to Z$ be functions. (a) Show that if $g \circ f$ is injective, then so is $f$. (b) If $g \circ f$ is surjective, must $g$ be surjective? ...
0
votes
1answer
25 views

Let T be a one-to-one linear transformation from $R^m$ to $R^n$ and B={$e_1$,$e_2$,…,$e_m$} a basis for $R^m$.

Prove that the set {T($e_1$),T($e_2$),...,T($e_m$)} is an independent set. Let T : $R^n$ → $R^m$ be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax for all x in ...
2
votes
2answers
34 views

Suppose $\{v_1,v_2,v_3\}$ is a basis for some subspace $V$ of $\mathbb R^m$.

Let $b$ be a vector in that subspace. Prove that if $b$ is orthogonal to all three basis vectors, then b has to be a zero vector. Hint: What is $\|b\|$ I do not know how to start this proof. Thanks ...
4
votes
3answers
439 views

Prove that the integers $x$, $x+6$, $x+12$, $x+18$, $x+24$ can only be prime if $x$ is $5$.

Prove that the integers $x$, $x+6$, $x+12$, $x+18$, $x+24$ can only be prime if $x$ is $5$. I am very new to proofs and not completely sure of how to approach this one. I tried several different ...
3
votes
3answers
58 views

Exhibit a bijective function $\Bbb Z \to \Bbb Z$ with infinitely many orbits

I've the following exercise: Give an example of a bijective function $\Bbb Z\rightarrow\Bbb Z$ with infinitely many orbits. What would be its infinite orbits?
1
vote
2answers
64 views

Josephine problem

So the problem is Suppose there are 2n people in a circle; the first n are “good guys” and the last n are “bad guys.” Show that there is always an integer m (depending on n) such that, if we go ...
1
vote
3answers
77 views

No continuous injective map $f: \mathbb{S}^1 \to \mathbb{R}$ [duplicate]

A friend asked me if there could be a continuous injective map $$f: \mathbb{S}^1 \to \mathbb{R}.$$ My intuition tells me no. Endow $\mathbb{S}^1$ with a topology $\mathscr{T}$ and fix a pole $x ...
1
vote
1answer
39 views

Injections from a set of functions to R

Show there is an injection from $\Bbb R^2 \to \Bbb R $ Does there exist an injection from $X \to \Bbb R$ where $X $ is the set of all functions where f(x)=x for all but finitely many x. This is a ...
6
votes
4answers
78 views

Prove A is an open set if and only if $A \cap Bd(A) = \emptyset $

Prove A is an open set if and only if $A \cap Bd(A) = \emptyset $ Here is my start: Suppose A is an open set. We know $X-A$ is closed. Need to show $A \cap Bd(A) = \emptyset$ Let $ x \in A$. ...
1
vote
2answers
112 views

How can I find $(x,y)$ so that $x^y=y^x$, $x\neq y$? [duplicate]

I have found that there are two $(x,y)$ that fulfill the property that $x^y=y^x$, $x\neq y$: $(2,4)$ $(4,2)$ From this: How can I find more, if any? How can I prove that there are no other ...
1
vote
2answers
55 views

Combinatorial Proof of falling factorial and binomial theorem

For $n,m,k \in \mathbb{N}$ is true: $$(n+m)^{\underline{k}}=\sum^{k}_{i=0}\binom ki \cdot n^{\underline{k-i}} \cdot m^{\underline{i}}$$ I can prove the binomial theorem for itself combinatorically ...
2
votes
2answers
53 views

Proof by Contradiction Minimum Value Proof $f(x)$

Focusing on $x=a$ first. My Proof: Assume $f'(a) < 0$ $f(x) \le f(x_1)$ for all $x$, this follows from the extreme value theorem. $$f'(x_1) = 0$$ Because it is a maximum. $$\exists x_4 ...
1
vote
1answer
32 views

Show that if $\lim_{x \to a}f(x)=L$ then $f$ is bounded near $a$.

The Problem: Show that if $\lim_{x \to a}f(x)=L$ then $f$ is bounded near $a$, i.e. there are constants $C,M > 0$ such that $\left|f(x)\right|<M$ for all $x$ such that $\left|x-a\right| < ...
0
votes
1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
1
vote
0answers
33 views

people passing a bridge (a proof for a greedy algorithm)

the problem some people are passing a bridge . each one takes a different time to pass . assume the people are sorted by their passing time increasingly . these are the conditions of passing the ...
4
votes
2answers
47 views

If $A \subsetneq C$ then $A \subsetneq B$ or $B \subsetneq C$. Contrapositive?

If $A \subsetneq C$ then $A \subsetneq B$ or $B \subsetneq C$. Is the contrapositive of this statement If $A \subseteq B$ AND $B \subseteq C$ then $A \subseteq C$. I asked because I think the ...
2
votes
2answers
70 views

Proving a Problem involving Fibonacci numbers

I'm working on proving the problem that states $\text {The sequence}$ {$F_n$} $\text {is defined by the} \ F_1=F_2=1, F_{n+2}=F_{n+1}+F_n \ \text {for} \ n \ge 1.$ $\text {For any natural number m, ...
1
vote
4answers
31 views

A={n∈ℤ: 6∣n and 8∣n} and B={n∈ℤ: 48∣n}. Is A⊆B?

A={n∈ℤ: 6∣n and 8∣n} and B={n∈ℤ: 48∣n}. Is A⊆B? Is B⊆A? I'm pretty sure that they are subsets of each other, because any n that 6 and 8 would both divide would have to be divisible by 6*8, but I'm ...
11
votes
5answers
683 views

Finite Abelian Group

Let $G$ be a finite abelian group, $G = \{e, a_{1}, a_{2}, ..., a_{n} \}$. Prove that $(a_{1}a_{2}\cdot \cdot \cdot a_{n})^{2}$ = $e$. I've been stuck on this problem for quite some time. Could ...
1
vote
3answers
40 views

Prove there exists $m$ and $k$ such that $ n = mk^2$ where $m$ is not a multiple of the square of any prime.

For any positive integer $n$, prove that there exists integers $m$ and $k$ such that: $$n = mk^2 $$ where $m$ is not a multiple of the square of any prime. (For all primes $p$, $p^2$ does not divide ...
2
votes
4answers
31 views

How to show a set of functions is or is not an open set on the sup-metric?

This is an excerpt from my text: The set $G$ of functions $g:\mathbb{R}\rightarrow\mathbb{R}$ such that $|g(x)|\le 1$ for all $x$ is not an open set in the sup-metric space. For instance, consider ...
3
votes
1answer
79 views

Satisfiability proof of formulas with pure literals

Let $\varphi$ be any propositional formula in negation normal form (NNF). A literal $\ell$ is pure in a formula $\varphi$, if the complement of $\ell$, $\ell^c$, does not occur in $\varphi$, where ...
0
votes
1answer
25 views

Proof of inverse of composite functions

Let $A$, $B$ & $C$ sets, and left $f:A \rightarrow B$ and $g:B \rightarrow C$ be functions. Suppose that $f$ and $g$ have inverses. Prove that $g\circ f$ has an inverse, and that $(g\circ ...
2
votes
1answer
50 views

Verification of Solution for Walter Rudin Principles of Mathematical Analysis Exercise 20, Chapter 3

I have written an answer for the problem 20, chapter 3 of Walter Rudin's Principle of Mathematical Analysis. I think the proof is correct, but since I am new with this kind of proofs, I am skeptical ...
1
vote
0answers
29 views

Finite dense subset implies $X$ finite

Suppose $E \subset X$ is a finite dense subset. Prove that $X$ must also be finite. This is proven quite easily by showing that $\bar{E} = E$ since $E' = \emptyset$, so that $\bar{E} = X$. ...
0
votes
1answer
23 views

Proof of Transitivity

Let R be the following relation of x and y on Z where 3x + y is even. I can seem to get to the form of $3x + z$ when I am doing algebraic manipulations if this equation. I have $3x + y = 2k$ and $3y ...
1
vote
1answer
69 views

Convergence of a sequence by convergence of sub-subsequence

Suppose that $\{p_n\}_{n \in \mathbb{N}}$ is a sequence in a metric space $X$. Assuming that every subsequence of $\{p_n\}_{n \in \mathbb{N}}$ has itself a subsequence that converges, say, to $p$, ...
2
votes
1answer
40 views

Proving consecutive quadratic residue modulo p [duplicate]

Let p be a prime with p > 7. Prove that there are at least two consecutive quadratic residues modulo p. [Hint: Think about what integers will always be quadratic residues modulo p when p ≥ 7.]
1
vote
1answer
45 views

How to prove that $I+A^{T}A$ is invertible [duplicate]

Let $A$ be any $m\times n$ matrix and $I$ be the $n\times n$ identity. Prove that $I+A^{T}A$ is invertible.
3
votes
2answers
55 views

Understanding a proof conceptually

Let's assume that $V$ and $W$ are vector spaces over a field $\mathbb{K}$, $\lambda\in\mathbb{K}$, $\lambda\neq0$. $S: V\rightarrow W$ and $T: W\rightarrow V$ are linear maps. Prove, that $\lambda$ ...
1
vote
3answers
45 views

Proving existence of a unique real number

I am working on the following question: For all $x \in \mathbb{R}$, $x \neq 6$, there exists a unique real number $y$ such that $xy+x=6y$. Now I have the existence part. That there exists a ...
0
votes
2answers
14 views

Proof of Injection and Surjection

I am having trouble proving the function f is injective and surjective. $f$ is a function from $\mathbb{Z}\times{Z} \to $\mathbb{Z}\times{Z}$ and $f(x,y) = (5x-y,x+y)$. I know it should be fairly ...
0
votes
2answers
19 views

Let $\text{A}$ be a nonsingular $\textit{n}\times\textit{n}$ matrix, and let $\textit{B}$ be a basis for $\mathbb{R}^n$

Show that $ B_1 = \{\textbf{Av}| \textbf{v} \in B\} $ is also a basis for $\mathbb{R}^n.$ I apologize for my informality, but I would really like some feedback as to whether I am using the correct ...
1
vote
1answer
36 views

Proving Inequality using Induction.

I am trying to prove the following statement: For every nonnegative integer $n$, $1+6n \le 7^n$. I did the base case where $n=0$ but am having trouble manipulating the inductive step. So far I ...
0
votes
3answers
84 views

If $ f $ is injective and $ g $ is injective, then $ f \circ g $ is surjective. [duplicate]

I can prove that if $ f $ and $ g $ are both injective, then $ f \circ g $ is injective, but I don’t know how to prove that $ f \circ g $ is surjective.
1
vote
0answers
67 views

Is the solution to this elementary number theory problem correct?

Problem: A natural number $n$ is called nice if the following properties hold: • The expression is made ​​up of 4 decimal digits; • the first and third digits of $n$ are equal; • the second and ...
1
vote
1answer
51 views

Let $A = \{1- \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$.

Let $A = \{1 - \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$. I am supposed to figure out if this set is closed under certain topologies. I know that means I ...
0
votes
2answers
22 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
1
vote
4answers
33 views

Monotonicity of a fraction.

So I want to prove that the following fraction is monotone increasing, as a part of another proof, that's why I stumbled on: $$\frac{4^{n+1}}{2\sqrt{n+1}} \ge \frac{4^{n}}{2\sqrt{n}}$$ I know it's ...
2
votes
2answers
71 views

Proving $\mathbb{Q} \times \mathbb{Z_2} \ncong \mathbb{Q}$

How do I prove $\mathbb{Q} \times \mathbb{Z_2} \ncong \mathbb{Q}$? I know that they are not isomorphic because for each element in $\mathbb{Q}$, say $\frac{a}{b}$, there are two corresponding elements ...
0
votes
2answers
25 views

proving a system with $A\vec{x} = \vec{0}=A^k\vec{x} $

The question: "Let $A\vec{x} = \vec{0}$ be a homogeneous system of $n$ linear equations in $n$ unknowns that has only the trivial solution. Show that if $k$ is any positive integer, then the ...
2
votes
2answers
32 views

Help with Factorial Inequality Induction proof

So I'm asked to prove $3^n + n! \le (n+3)!$, $\forall \ n \in \mathbb N$ by induction. However I'm getting stuck in the induction step. What I have is: (n=1) $3^{(1)} + (1)! = 4$ and $((1) + 3)! = ...
0
votes
1answer
26 views

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$.

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$. My definition of closure is "Let $(X, \mathfrak T)$ be ...
0
votes
0answers
21 views

Prove that h:$\mathbb{Z}\rightarrow\mathbb{O}$ where h(n)=2n-1 is bijective

I need to prove that h:$\mathbb{Z}\rightarrow\mathbb{O}$ where h(n)=2n-1 is bijective. I haven't done problems where $\mathbb{Z}\rightarrow\mathbb{O}$ and have seen no examples. I am only familiar ...
0
votes
1answer
52 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then A =∅ or $A = X$ .

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then $A = \emptyset$ or $A = X$. I am studying introduction to proofs and we have learned ...