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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2
votes
3answers
110 views

is it wrong to do this to solve an induction question

When doing an induction problem is it wrong to simply add the next variable to both sides? for example for all natural numbers $$4+9+14+19....+(5n-1)=\frac{n}{2}(3+5n)$$ assume true for k ...
2
votes
0answers
46 views

the sum of the first n odd integers squared proof [duplicate]

I don't know why but I can't even get my base case to work with this proof can someone help me out? Prove that $$1^2+3^2+5^2+\cdots+(2n+1)^2=\frac{(n+1)(2n+1)(2n+3)}{3}$$ when $n$ is a non negative ...
2
votes
3answers
62 views

Proof expectation of bernoulli distribution

Suppose we have: $P(X=k) = (1-p)^k p$ $$E(X) = \sum^{\infty}_{k=0} kP(X=k)= \sum^{\infty}_{k=0} kp(1-p)^k = p(1-p) \frac{1}{p^2}=\frac{1-p}{p}$$ What I do not get is the step in the equation ...
6
votes
5answers
361 views

Proof Strategy - Prove that each eigenvalue of $A^{2}$ is real and is less than or equal to zero - 2011 8C

Remember that we've already proven the following, for any real symmetric $n\times n$ matrix $M$: (i) Each eigenvalue of $M$ is real. (ii) Each eigenvector can be chosen to be real. (iii) Eigenvectors ...
0
votes
5answers
39 views

help solving this proof with remainders

For all $n\ge3\in \mathbb N$, if $n \equiv 3 \pmod{4}$ then ${3^n} \equiv 2 \pmod{5}$. I tried to set $n = 3+4k$ but it doesn't help. Any hints first please?
1
vote
5answers
126 views

How to prove that for all natural numbers, $4^n > n^3$?

This is a problem set I have, it's not a homework but it's very important practice... Send me some hints please, I don't want an answer I need to get it by myself but I'm failing miserably... The ...
1
vote
3answers
72 views

Proving if $\gcd(c,m)=1$ then $\{x\in \Bbb Z \mid ax\equiv b \pmod m\} =\{x\in \Bbb Z \mid cax\equiv cb \pmod m\}$

Okay so I'm confused on how to approach this question. If $\gcd(c,m)=1$, then $S=T$ where $S=\{x\in \Bbb Z \mid ax\equiv b \pmod m\}$ and $T=\{x\in \Bbb Z \mid cax\equiv cb \pmod m\}$. I know ...
0
votes
6answers
103 views

Suppose that $x$ and $y$ satisfy $\frac{x}{2} + \frac{y}{3} = 1$. Prove that $x^2 + y^2 > 1$.

Ok , i tried to prove this via Contrapositive setting $x^2 + y^2 \le 1$. After doing some algebra i have arrived at $x \le \sqrt{-y^2}$. I'm fairly sure this isn't right. I also solved for x and y in ...
1
vote
1answer
63 views

Proof by contradiction: $(A \subset \Bbb{R}) \land (A \neq \emptyset )\wedge (A \mbox{ is bounded below } )\to \exists x (x \doteq \inf(A))$

I must proof the following: Prop.: let be $\Bbb{R}$ a complete ordered field $$\emptyset \neq A \subset \Bbb{R} \wedge A \mbox{ is bounded below } \to \exists x (x \doteq \inf(A))$$ Proof: by ...
1
vote
1answer
136 views

Suppose that 0 < a < b. Prove that $a < \sqrt{ab} < b $ and $\sqrt{ab} \leq \frac{1}{2}(a+b)$

For part 1, I have used the NOT operator on it, giving me $a \geq \sqrt{ab} \geq b$, and then tried to prove a contradiction to the assumption. I came up with $a = b$ by transitivity, which ...
4
votes
3answers
157 views

Proof: $ \lfloor \sqrt{ \lfloor x\rfloor } \rfloor = \lfloor\sqrt{x}\rfloor $.

I need some help with the following proof: $ \lfloor \sqrt{ \lfloor x\rfloor } \rfloor = \lfloor\sqrt{x}\rfloor $. I got: (1) $[ \sqrt{x} ] \le \sqrt{x} < [\sqrt{x}] + 1 $ (by definition?). (2) ...
0
votes
0answers
29 views

Shorten a proof using Galois connections

Consider a Galois connection: $f:R\rightarrow F$ is a lower adjoint of $r:F\rightarrow R$ for partially ordered sets (actually complete lattices) $F$ and $R$. We have also $f(r(g))=g$ for every $g\in ...
2
votes
1answer
35 views

Is it possible to infer this relation without calculation?

Suppose $A\sim \mathscr{E}(\alpha)$ and $B\sim\mathscr{E}(\beta)$. Is it possible to argue that: $$\beta\,\mathbb{P}(A>B)=\alpha\,\mathbb{P}(B>A)$$ without calculating $\mathbb{P}(A>B)$ or ...
1
vote
1answer
57 views

How to presage Prove by Contrapositive, for Sequential Characterizations of Limit and Continuity? (Abbott pp 106 t4.2.3, 110 t4.3.2)

Dafinguzman answered consummately this question initially but it became too long. I want to question for different beliefs. 1. $(ii) \implies (i)$ in both Theorems 4.12 and 4.19 posit sequences ...
3
votes
3answers
418 views

Prove: Number of Derangement is odd if and only if number of items is even .

let $D_n$ be a number of Derangement of n items . prove that $D_n$ is odd if and only if n is even . i was trying to use induction on the $!n=(n-1)(!(n-1)+!(n-2))$ recurrence relation but i cant ...
11
votes
2answers
1k views

Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
0
votes
2answers
47 views

Basic proof of statement in abstract algebra?

http://www.proofwiki.org/wiki/Abelian_Quotient_Group The third step (in both proofs) is something I am having trouble seeing. The theorem itself is not difficult to prove, but it is much cleaner this ...
0
votes
1answer
53 views

Struggling to prove that if $n$ is a non zero integer, and $m > 0 \mid n$ then $m \leq |n|$

i need to prove that if $n$ is a non zero integer, and $m > 0$ and $m \mid n$ ($m$ divides $n$), then $m \le |n|$. I feel like i can do it by a combination of proof by contradiction and cases (ie ...
2
votes
1answer
62 views

Proof: $A=\{c_1,c_2,…,c_n\}\subset \Bbb{R} \to A \mbox{ is bounded above }$

I must proof the following: Prop.: Let be $\Bbb{R}$ a complete ordered field $$A=\{c_1,c_2,...,c_n\}\subset \Bbb{R} \to A \mbox{ is bounded above }$$ Proof.: by induction on $n$, with $n \geq 1$ ...
0
votes
1answer
76 views

Prove or Disprove the following statement. For any sets $A$, $ B$, and $C$, we have $A \cup (B \& C) = (A\cup B) \cup (A\&C)$

Trying to figure this question out in my proofs class (tried venn-diagram the multiple set-notation signs are confusing me). Homework question in the fundamental sets unit.
-2
votes
1answer
56 views

Let ~ be an equivalence relation on a set S. Show that b is an element of cl(a) <=> cl(a) = cl(b) (Where all a,b are elements of S)

This was a question on my last equivalence relations quiz and I'm not yet comfortable with the whole "class" idea. I understand that I must show transitivity, reflexivity and symmetry however I'm not ...
0
votes
2answers
301 views

Determine the number of equivalence relations on the set {1, 2, 3, 4}

Hi this was a question listed on my last proofs and conjectures midterm. It is similar to my previous post however this asks a different question which is throwing me off.. Do I simply list all ...
0
votes
2answers
54 views

Determining whether relations are equivalence classes, and finding the equivalence classes

Determine if each of the following relations is an equivalence relation. If so, determine the equivalence classes. $S = \Bbb Z$, $a \sim b \iff a \equiv b \pmod 3$ or $a \equiv b \pmod 5$. ...
2
votes
3answers
113 views

Prove $\det(A)=\det(A^T)$ detail

I want to prove that $$\det(A)=\det(A^T)$$ and the one step I don't understand (the problem is guiding you thought it is to prove $$P^T_{\sigma} = P_{\sigma^{-1}}$$ where P is a permutation matrix. ...
1
vote
0answers
86 views

Intuition for $\inf(AB) = \inf(A)\sup(B)$. Difference for sets and functions? (Abbott pp 199 q7.4.5)

1. What's the intuition for $\inf(AB) = \inf(A)\sup(B)$? Figure please? I know I must posit $A,B \subseteq R$ as bounded sets. If they're unbounded, $\sup$ doesn't exist. I believe $\inf(AB) = ...
1
vote
1answer
51 views

Positive sinus on given points

I have N points: $x_1, \dots, x_N $ How to proof that exist $\alpha$ such that $\sin (\alpha x_i) > 0 \quad \forall i \ \in 1... N$
1
vote
0answers
89 views

So we don't need to choose delta, epsilon, or $N \in \mathbb{N}$ in delta-epsilon or sequence convergence proofs?

(http://math.stackexchange.com/a/700667/85079) I would write the proof with all my bounds $\eta$ and then choose $\eta$ to make the conclusion match the arbitrary $\epsilon$. ...
1
vote
1answer
75 views

If $g \ge 0$ is continuous on $[a,b]$ and $g(x_0) > 0$ then $\int^{b}_a g > 0$ (Abbott pp 199 q7.4.4c)

True or False. If $g \ge 0$ is continuous on $[a,b]$ and $g(x_0) > 0$ for $\ge 1$ point $x_0 \in [a,b]$, then $\int^{b}_a g > 0.$ 1. Need determine if true or false. Ergo do we need ...
1
vote
3answers
101 views

If $\int^{b}_a f > 0$ then there is some interval and $\delta > 0$ on which $f(x) \ge \delta$ (Abbott pp 199 q7.4.4d)

True or False. If $\int^{b}_a f > 0$, then $\exists \; [c,d] \subseteq [a,b]$ and $\delta > 0$ such that $f(x) \ge \delta$ for all $x \in [c,d]$. 1. We need to determine if true or false. ...
3
votes
1answer
204 views

How do you get good at reading research papers with lots of proofs?

tl;dr To get good at math proofs (and thinking math), do you have to first memorize all the different proof tricks, or is there a way to learn as you go? I am a software developer and have recently ...
1
vote
2answers
74 views

Why does $\lim_{n \to\infty}a_{n+1} = \lim_{n\to\infty}a_n$?

Assume that $\{a_n\}$ is a convergent sequence. How to use the definition of a limit of a sequence to prove that
-1
votes
1answer
39 views

Prove: $a < a^n$ (more details in description)

Let $\rm\:a\in \mathbb Z.\:$ Prove that if $\rm\: a > 1,$ then for all $\rm\:n > 1, a < a^n.$
0
votes
1answer
72 views

Differential map on the vector space of polynomials: Kernel and Image

Given the $V_n$ is the vector space of polynomials of degree $\leq$ n over $\Bbb R$ So $M_D = \begin{bmatrix} 0 & 1 & ... & 0\\ 0 & 0 & 2 &... 0 \\ 0 & 0 &0 &.\\ . ...
2
votes
1answer
90 views

Proof. sup{ f(x) } - inf{ f(x) } $\ge$ sup{ |f(x)| } - inf{ |f(x)| } (Abbott pp 198 q7.4.1)

Let f be a bounded function on a set A, and set $S = \sup\{f(x) : x ∈ A\}, I = \inf \{f(x) : x ∈ A\},$ $S' = \sup\{|f(x)| : x ∈ A\}, I' = \inf \{|f(x)| : x ∈ A\}.$ Show that $S - I ≥ S' - I'$. ...
3
votes
3answers
109 views

When does one proof of one direction of an If and Only If proof suffice?

Would someone please explain when this is admissible (please expound on $\color{darkred}{sometimes}$)? In advance of starting an Iff proof, how would one divine/previse if this convenience (of a ...
8
votes
2answers
141 views

Prove $ \int_{C}fdr=\int_{S}dS\times\nabla f$

Prove $\displaystyle \int_{C}fdr=\int_{S}dS\times\nabla f$. where $C=\partial S$ and the usual relationship between orientations hold. Apply Stokes's theorem to $F=af$ where $a$ is an arbitrary ...
-1
votes
1answer
67 views

Looking for a lemma and its proof

I recently acquired this following lemma for general topology and would like to know where (i.e. book) this was from, also if possible, a proof for it. Does this look familiar to anyone? Lemma: Let ...
0
votes
1answer
17 views

Proving the existence of a ____ number of conditions under a rotation of coordinates.

I'm reading a section on Rotation of Coordinate Systems and this is throwing me off: 'In n-dimensional space, the rotation matrix will have $n^2$ elements, upon which orthogonality relations place ...
1
vote
2answers
39 views

Prove by induction on a string

I want to practice proving the following: Given a binary string s, I want to prove $s$ has the same number of substrings 01 and 10 $\iff$ the first and last character of $s$ is the same. For ...
2
votes
1answer
50 views

The rational unit distance graph is bipartite

I am trying questions from a Graph theory book by Bondy and Murty. I stumbled across a neat looking problem. The unit distance graph on a subset $V$ of $\mathbb{R}^2$ is the graph with vertex set ...
3
votes
3answers
144 views

Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$

Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n}) $ I tried induction theorem, when $n = 1$ it is obviously right. But, say $n=k$, It does not make sense since I cannot ...
1
vote
2answers
30 views

Executing a proof given a homeomorphism

The question I am proving: If $f: X \to Y$ is a homeomorphism show that $A$ is a closed subset of $X$ if and only if $f\left(A\right)$ is a closed subset of $Y$. The proof that I want to use mimicks ...
1
vote
1answer
19 views

Every free module is a projective one

I'm trying to understand this proof in Hungerford's book using the universal property of the free modules: In the whole proof I didn't understand just this line, because we can use the uniqueness ...
1
vote
1answer
65 views

Intuition. Cauchy criteron for Riemann integrability (Spivak pp 239, S. Abbott pp 189 thm 7.2.8)

1. Why $\inf U(f,P') \le U(f, P)$ and $\sup L(f, P') \ge L(f,P) $? I tried to research but I can't find where Spivak defined it $P'$? 2. Why are there two partitions P', P''? Not the same? ...
1
vote
1answer
69 views

Proof of Extreme Value Theorem. Modus operandi. (S. Abbott pp 115 t4.4.3)

By pp 115 Abbott Theorem 4.4.2, we know that $f([a, b])$ is bounded above and below. $f([a, b])$ contains [a,b] ergo it is clearly nonempty). By the agency of the Axiom of Completeness, it has a ...
0
votes
2answers
23 views

proof by induction - creating summations?

I have two proofs I need to do that I can not figure out how to turn into summations in order to solve. $3|(4^n-1)$ I believe that $|$ is meant to symbolize $3$ divides ... $n!\le n^n$ I have to ...
0
votes
4answers
60 views

if x^2 + 2x - 3 >= 0 then (x <= -3) V (x >= 1)

I know why this is true but putting it in symbolic notation has me stumped. so basically i have that: ...
1
vote
1answer
54 views

subgroup proof.

Prove that if $G$ is an abelian group, then $H =\{ x \in G\mid x^{2} = e \}$ is a subgroup of $G$. I did show that $H$ is close, associative, have identity and inverse element. Then my prof said I ...
2
votes
1answer
43 views

Definition of $Z_m$ is $[n] = \{x | x \equiv n \pmod m\}$?

Any help or sort of input on this question would help a great deal. Thanks Let $m\in N$. Recall for any integer $n \in Z$, the definition of $[n]$ in $Z_m$ is $[n] = \{x | x \equiv n \pmod m\}$. ...
1
vote
2answers
271 views

Proof: Subsequence of n integers is divisible by n?

So I'm confused and stuck on how to approach this question. Any direction in the right path would be greatly appreciated. Let $n\in N$. Prove that any sequence of $n$ integers $a_1, a_2, ... a_n$ (no ...