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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0answers
34 views

Proof of independence (linear algebra) [duplicate]

Prove that {1,sin(x),sin(2x),sin(3x),…,sin(nx)} is an independent set. I received 1 answer 3 days ago but didn't find the answer very clear. Please advice, thanks!
1
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2answers
85 views

Proof $(P \land S) \rightarrow \lnot q$ in hilbert system

How can I proof $(P \land S) \rightarrow \lnot q$ using this principles : $p \rightarrow (q \lor r) $ $q \rightarrow \lnot r $ $r \leftrightarrow s $ in Hilbert system and modus ponens?
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1answer
85 views

Prove $\frac {1}{\cos 0^\circ \cdot \cos 1^\circ} + \ldots +\frac {1}{\cos 88^\circ \cdot \cos 89^\circ}= \frac{\cos 1^\circ}{\sin 1^\circ}$

Prove the following identity: $$\frac {1}{\cos 0^{\circ} \cdot \cos 1^{\circ}} + \ldots +\frac {1}{\cos 88^{\circ} \cdot \cos 89^{\circ}} = \frac{\cos 1^{\circ}}{\sin 1^{\circ}}$$ After hours of ...
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2answers
332 views

Prove, for any positive integer $n$, that $n -3$ must be a multiple of $5$ if $n^3 -n -4$ is a multiple of $5$.

I had previously solved the problem of proving that $n^3-n-4$ must be a multiple of $5$, given that $n-3$ is a multiple of $5$. I did so by algebraically manipulating $n^3-n-4$ into: $$ ...
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1answer
22 views

Proof with Segments

I have an assignment (from a tutor) that tells me: Give an informal proof for "If A-B-C and point P is on segment AC, then P is on segment AB or P is on segment BC" To start my proof, I have assumed ...
2
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1answer
52 views

Which functions are 1-1 and which are onto?

Let $N$ denote the naturals, $Z$ the integers, $R$ the reals. Which of the following is 1-1? Onto? $(a) f(x) = x^2$ on $N, Z, R$ Suppose $f(x_1), f(x_2) \in N$. Since $x_1, x_2 \in N : x_1 \neq ...
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1answer
148 views

For every $\epsilon >0$ , if $a+\epsilon >b$ , can we conclude $a>b$?

If $a+\epsilon > b$ for each $\epsilon >0$, can we conclude that $a>b$? Please help me to clarify the above. Thanks in advance.
0
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1answer
66 views

Proving that two crossbars of a bisector intersect the midpoint of one of the crossbars

This is sort of an involved question. I've proved parts of it already but now I'm stuck. Here is the question: Let ∆ABC be such that AB is not congruent to AC. Let D be the point of intersection of ...
3
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2answers
63 views

Inequality Proof, need help simplifying

I'm new here and unsure if this is the right way to format a problem, but here goes nothing. I'm currently trying to solve an inequality proof to show that $n^3 > 2n+1$ for all $n \geq 2$. I ...
2
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1answer
169 views

Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$

I want to verify the proof of this result and get some start ideas to overcome the different steps of this proof. Lemma: Let $g$ be a real analytic function. Then we have the equivalence ...
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2answers
49 views

Prove that a number is not a square root

Prove, if $n \in \mathbb{N}, b \in \mathbb{Z}$ then $\left( 6n+2\right) \neq b ^{2}$. Just give me a hint. I've been trying to solve it for over na hour.
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3answers
82 views

How to prove $(1+x)^n\geq 1+nx+\frac{n(n-1)}{2}x^2$ for all $x\geq 0$ and $n\geq 1$?

I've got most of the inductive work done but I'm stuck near the very end. I'm not so great with using induction when inequalities are involved, so I have no idea how to get what I need... ...
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2answers
97 views

A case of Bolzano-Weierstrass

Can anyone help me get started on this problem? Suppose {$a_n$} is a bounded sequence. Let $L$ = sup{$a_n| n\in \mathbb{N}$}. a) Prove that if $L\notin$ {$a_n| n\in \mathbb{N}$} then there is a ...
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2answers
404 views

Proof of the limit laws (Analysis)

Hi everyone I'd like to know if my arguments for the next proof are sound or needs some changes to be correct. I hope they are not a little flaws. Proposition (limit laws): Let $(a_n)_{n=m}^\infty$ ...
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2answers
60 views

Can I do the same things on both sides when I have a not equal sign?

I am trying to prove this statement: If $A$ is not symmetric then $A^{-1}$ is not symmetric And the following is my proof If $A≠A^T$, \begin{align*} & A^{-1}A=I≠A^{-1} A^T \\ & I^T \neq ...
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1answer
208 views
1
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2answers
199 views

Find the integration of $\sec(x)$ and prove it

My hw told me to prove the integral of $\sin(x)$, $\cos(x)$, $\tan(x)$, but when I get to $\sec(x)$ I'm stuck. I can find a way to prove it. Please help on explaining the integral of $\sec(x)$. ...
0
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2answers
124 views

Transitivity of parallel lines

I cam across a question (in my textbook) about proofs with parallel lines. The question is: Prove that the property that || is transitive implies that for any point P and line l, there is at the most ...
0
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1answer
54 views

Linearly Independent list of length 2

A list $S$ of length $2$ is linearly independent if and only if neither vector is a scalar multiple of the other. I'm not entirely sure precisely how to show this, but here are my thoughts: Let ...
1
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2answers
51 views

If $n = mp^e$ where $e$ is maximal, then $\binom{n}{p^e}$ is not divisible by $p$.

Let $n \geq 2$ be an integer, $p$ a prime with $p^e$ the highest power of $p$ dividing $n$. Then $\binom{n}{p^e}$ is not divisible by $p$. I think you can do it using this formula for ...
2
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5answers
242 views

Prove that a function does not have a limit as $x\rightarrow \infty $

Problem statement: Prove that the function $f(x)=\sin x$ does not have a limit as $x\rightarrow \infty $. Progress: I want to construct a $\varepsilon -\delta $-proof of this so first begin by ...
1
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2answers
101 views

Proof that this is independent

Prove that {$1, \sin(x), \sin(2x), \sin(3x),\ldots, \sin(nx)$} is an independent set. I can think of the long way which is to differentiate this and put the differentiations into a matrix and row ...
2
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1answer
58 views

Why are strict inequalities stronger than non-strict inequalities?

I'm working with induction proofs involving inequalities and I am encountering example proofs that wish to show things of the sort, $n!\le\ n^n$ for every positive integer. The proof given in the ...
1
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1answer
122 views

Proof by contradiction by first assuming proposition true?

In a proof by contradiction, we first assume a proposition $P$ false, then prove some known truth to be false, then that would imply the assumption $P$ should really be true. Do we really need to ...
1
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3answers
165 views

Mathematical induction - what makes it true?

I am trying to work through an example in my book and it just seems nonsensical Why is mathematical induction a valid proof technique? The reason comes from the well ordering property, listed in ...
1
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1answer
151 views

if x≡ a (mod m), x ≡ a (mod n) then, n|m

x≡ a (mod m), x ≡ a (mod n) if only if then, n|m. Original question is let a an integer, prove [a]_n is a subset of [a]_m if only if n|m. I wanted to prove it by choosing an element. So for the ...
3
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1answer
101 views

Proof for recursively defined sets

Language $L\subset \{a,b\}^*$ is such that: $\epsilon \in L$ $a \in L$ For any $x\in L$, $xb\in L$ and $xba\in L$ Nothing else in $L$. Im just learning recursive sets, but with that definition am ...
1
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2answers
211 views

Can mathematical induction be used to disprove something?

I saw this to be the rule of inference for mathematical induction : Now consider : as L.H.S. and as R.H.S.. Now if suppose, while trying to prove P(k) -> P(k+1), in the left hand side of ...
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1answer
1k views

Proving/Disproving Product of two irrational number is irrational

I saw this question where I had to prove/disprove that: Ques. Product of two irrational number is irrational. I tried 'Proof by Contraposition'. Product of two irrational number is irrational. p ...
0
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1answer
629 views

Proof for Symmetry property of Congruent Segments

I am starting to learn geometrical proofs, and I have come across the Symmetry property of segment congruence (if $AB$ is congruent to $CD$, then $CD$ is congruent to $AB$). One of the exercises in ...
5
votes
1answer
49 views

Is this a valid proof of the contrapositive?

The question is the following: if $a$ and $b$ are distinct group elements, then either $a^2 \neq b^2$ or $a^3 \neq b^3$. I find this difficult to prove directly, so I formulated the contrapositive to ...
1
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1answer
45 views

Equivalence of different definitions of “monotonically normal space.”

Definitions/Background: A monotone normality operator on a topological space $X$ is a function $\mathcal G$ such that for any two closed disjoint subsets $E,F$ of $X,$ $\mathcal G(E,F)$ is an open ...
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2answers
79 views

Using the definition of sqrt(x)

I am having trouble understanding what the definition of $\sqrt{x}$ is. I guess it's $x = a^2$ where $a$ is an integer. The question asks for: Use the exact definition of $\sqrt{x}$ to prove that $x ...
2
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1answer
115 views

Help with a lemma of the nth root (without the binomial formula)

I have no idea of how to solve it. I would appreciate if someone gives me a hint, please. Definitions Let $\,x^{1/n}:= sup\{\, y \in \mathbb{R}: y\ge0 \text{ and } y^n\le x\, \}$ Lemma: Let ...
4
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2answers
239 views

If $x$ and $y$ are rational then is $x^y$ also rational?

I can think of the counter example $x = 2$ and $y = 1/2$ but how would a proof to disprove this look like?
3
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2answers
67 views

Prove through induction that $3^n > n^3$ for $n \geq 4$

I'm new to induction and have not done induction with inequalities before, so I get stuck at proving after the 3rd step. The question is: Use induction to show that $3^n > n^3$ for $n \geq ...
1
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1answer
188 views

Proof verification on Fermat's Little Theorem exercise - new way to solve problem?

I don't know if I'm correct, since I didn't even have to use the hint. So I'm asking for proof verification since I am also not too confident on primes. Suppose $\gcd(a, 35) = 1.$ Show that ...
1
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0answers
57 views

Is this FLT number theory proof correct?

Let $p = 5$. Verify that $p | 14^4 - 1$ but not $15^4 - 1$. Does the latter contradict Fermat's Little Theorem? Suppose $5 | 14^4 - 1$. Then $14^4 \equiv 1 mod (5)$. Since $gcd(5, 14) = 1$, then ...
0
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1answer
106 views

How do I prove an algorithm has $n^3$ time complexity?

Take the CYK algorithm outlined here: How to prove CYK algorithm has $O(n^3)$ running time In the top answer, how did that person go from the three summations to $t=(n^3−n)/6$ ? What's the method ...
0
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1answer
207 views

Need help understanding, proving that for every natural number n, f(n) is the largest power of 2 less than or equal to n.

$$f(1) = 1.$$ for $n\geq$1; $$f(2n) = f(2n + 1) = 2f(n).$$ I don't understand what is going on. Am I asked to prove that $f(n) = 2^{k}$ and $k\leq n$?
0
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1answer
54 views

Prove that projection of f onto g is orthogonal to f minus projection of f onto g.

Using basic properties of the integral and the defintions of norm, dot product, orthogonality, and projection given above, show that, for any two continuous functions f and g defined on [0; 1], ...
0
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1answer
48 views

Is simple straight-edge and compass construction a substantial proof?

I'm working on a problem that asks to prove that a point $D$ is outside of a $\triangle ABC$, on the circle through the triangle, given that sides $AB$ and $AC$ are not congruent, and that $D$ is the ...
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1answer
59 views

Geometry Proof Triangles

Show that if two of the corresponding angles of two triangles are equal then so is the third. Is there a formal way to prove this? I wanted to just say in one sentence that if two angles are the ...
1
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0answers
22 views

Chi-squared test property

I have got the following: $$\sum_{i=1}^{k-1} \frac{O_i - N p_i }{N p_i + (1-p_i)} \sim \chi^2_{k-1} $$ How to prove that $$\sum_{i=1}^{k} \frac{O_i - N p_i }{N p_i} \sim \chi^2_{k-1} $$? Where: ...
0
votes
1answer
62 views

Max function with probabilities

I have the following: $$p(Y<y) = p(\max(x_1, x_2, \ldots, x_t) < y)$$ Where $x_1, x_2, \ldots, x_t$ are independent(they come from a sample) why the following is true? $$p(\max(x_1, x_2, ...
1
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2answers
48 views

Does $|\textbf{x}-\textbf y|<\delta$ imply $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$

I want to say that $|\textbf{x}-\textbf y|<\delta$ implies $|x_1- y_1|<\delta$ and $|x_2- y_2|<\delta$ for a proof I am working on. This is assuming that $\textbf{x}=(x_1,x_2) \in \text R^2$ ...
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2answers
105 views

Every positive rational number $x$ can be expressed in the form $\sum_{k=1}^n \frac{a_k}{k!}$ with $ a_k≤ k − 1$ for $k ≥ 2$ .

I have this theorem which I can't prove.Please help. "Show that every positive rational number $x$ can be expressed in the form $\sum_{k=1}^n \frac{a_k}{k!}$ in one and only one way where each $a_k$ ...
0
votes
1answer
47 views

Question about $\mathbb{Z}_n$ and $G_n$

For $n ≥ 2$ we let $G_n ⊂ Z_n$ denote the subset of all integers mod n which are invertible $\mod n$. Let $m, n \in \mathbb{Z}$ $m, n ≥ 2$ and $(m, n) = 1$. Define a mapping $f : Z_m × Z_n → Z_{mn}$ ...
1
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1answer
59 views

How to prove this inequality? $ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $

If $z$ is any non-zero complex number, how to prove the following inequality? $$ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $$ Hints please!
4
votes
1answer
219 views

Proof: Sequence of n consecutive natural numbers containing no primes (Velleman P158 Thm 3.7.3)

Theorem: For every positive integer $n$, there is a sequence of $n$ consecutive positive integers containing no primes. (Another MSE post about this Theorem) Proof: Since we desire "a sequence ...