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
6 views

About Mellin transform and harmonic series

Let $\mathfrak{M}\left(*\right)$ the Mellin transform. We know that holds this identity$$\mathfrak{M}\left(\underset{k\geq1}{\sum}\lambda_{k}g\left(\mu_{k}x\right),\, ...
0
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1answer
13 views

Proof of floor function identity.

Let $f(x) = \lfloor x \rfloor$ and let $l$ be the greatest integer $\le x$ How do I prove $l + 1 > x$ I see that: $x \ge \lfloor x \rfloor = l$ No complete answers, just hints
2
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1answer
10 views

Boolean Algebra: Converting $xy'z + wxy'z' + wxy + w'x'y'z' + w'x'yz' = w'x'z' + xy'z + wx$

Notation w,x,y,z are all just primary statements "+" is the OR logical operator what looks like two or more statements being multiplied is actually the AND operator The complement or prime ...
2
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1answer
22 views

Finding a specific term in a repeating sequence.

Let $f(x) = \frac1{1-x}$, and define the function $f^r$ by $$f^r(x):=\underbrace{f(f(f(...f(f}_{r\text{ times}}(x))))).$$ I am asked to find to find $f^{653} (56)$. I know that there are only $3$ ...
3
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4answers
243 views

How can I prove this integral is equal to f(0)?

Given that $f$ continuous over $[-1,1]$, how can I show $\lim_{x \to 0}\frac{1}{x}\int_0^xf(t)dt = f(0)$? I know the limit of $\frac{1}{X}$ doesn't exist at 0, and it's negative infinity from the ...
1
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2answers
31 views

Proving Pascal's identity

So I came across Pascal's identity: Prove that for any fixed $r\geq 1$, and all $n\geq r$, $$ \binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1}. $$ I know you can use basic algebra or even an inductive ...
2
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3answers
46 views

Prove that $0 < 1$. Prove that $ab = 0 \implies a = 0$ or $b = 0$.

Proof: There exists $a = 0$ (For every $b$, an element of the set of positive numbers, such that: $b > a$) $$a + b > 0 \implies b > 0 \implies a < b.$$ Thus, we have shown that $0 < ...
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2answers
24 views

Binomial coefficient identity $\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$ [duplicate]

I'm having a bit of problems proving the following: $$\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$$ I always seem to get to the line: $2^{n-1} + 1 = 2^n$ which I know is untrue. Could anyone ...
6
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3answers
69 views

Show $7!^{1/7} < 8!^{1/8}$

Show $7!^{1/7} < 8!^{1/8}$ So I know that the first step is to remove the radicals. So would I raise both sides to the power of 8 to get $({7!}^{1/7})^8 < 8!$. I am not sure where to go from ...
2
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2answers
49 views

Show that $a^2 + b^2 + c^2 \geq ab + bc + ca$ for all positive integers $a$, $b$, and $c$ [duplicate]

Show that $a^2 + b^2 + c^2 \geq ab + bc + ca$ for all positive integers $a$, $b$, and $c$. I am not sure how to approach this problem. Should I divide this problem into multiple cases based on ...
2
votes
2answers
40 views

What is the modus ponens of a tautology?

In the statement $P$ and $Q$, please show that $\; (P \land (P \Rightarrow Q))\Rightarrow Q \;$ is a tauntology. The state the $\;(P \land (P \Rightarrow Q))\Rightarrow Q\;$ in words. I know I need ...
2
votes
3answers
25 views

Prove that $\sqrt x$ is Lipschitz on $[1, \infty)$

Prove that $\sqrt x$ is Lipschitz on $[1, \infty)$ I want to show that $|f(x) - f(y)| \leq L |x - y|$ So $|\sqrt x - \sqrt y| = \frac{|x - y|}{\sqrt x + \sqrt y} \leq \frac{1}{2}|x - y|$. I can ...
1
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2answers
24 views

Proving if $A$ or $B$ are symmetric then $AB$ is symmetric

Prove if $A$ or $B$ are symmetric then $AB$ is symmetric. Symmetric set definition: $A$ is symmetric if for every $a\in A$ there's $-a\in A$ Product set definition: $AB=\{ab\mid a\in A, ...
2
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2answers
53 views

Is there another way to prove $(x-n)^2 = (n-x)^2$

Let's say $n$ is $4$. So, I came up with the solution below. $(x-4)^2 = (x-4)(x-4) = x^2 - 8x + 16$ $(4-x)^2 = (4-x)(4-x) = 16 - 8x + x^2 = x^2 - 8x + 16$ I was wondering if there is another way ...
0
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1answer
26 views

Rectangles in one dimension

I have to prove the following proposition : Show that the intesection of two rectangles in $\mathbb{R}^{n}$ is either the vaccum or is another rectangle. My attempt: I one is embeded in the other ...
1
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2answers
40 views

Proving if $|A|\ge 4 \vee |A|\le 2$ then $|A+A|\neq 4$ with direct, contradiction and contraposition

Prove if $|A|\ge 4 \vee |A|\le 2$ then $|A+A|\neq 4$. $A$ is some set and we define $A+B=\{a+b|a\in A, b\in B\}$, $A$ is some subset of the reals. In a direct proof and proof by contradiction I'd ...
1
vote
1answer
35 views

Using induction to show associativity on $x_1+\dots + x_n$

I want to use induction to show that the sum $x_1 + \dots + x_n$ of real numbers is defined independently of parentheses to specify order of addition. I know how to apply induction(base, assumption, ...
0
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0answers
26 views

How to prove that the integral of a positive, continuous function is positive?

Obviously intuitively the area under something that is above the x-axis is always positive, but how can I show this with a proof?
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5answers
49 views

Number theory proof [on hold]

$(i)$ Prove that for every natural number $n \geq 2$, one has $(n + 1)|(n^3 + 1)$; $(ii)$ Suppose that $n$ is a natural number exceeding $1$. Prove that $(n^2-1)|(n^3+1)$ if and only if $n = 2$.
2
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2answers
48 views

Proving with induction $(1-x)^n<\frac 1 {1+nx}$

Prove using induction that $\forall n\in\mathbb N, \forall x\in \mathbb R: 0<x<1: (1-x)^n<\frac 1 {1+nx}$ My attempt: Base: for $n=1: 1-x<\frac 1 {1+x}\iff 1-x^2<1$, true since ...
0
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1answer
30 views

prove that $p^2-1$ is divisible by $24$ if $p$ is a prime greater than $3$ [duplicate]

How to prove that $p^2-1$ is divisible by $24$ if $p$ is a prime number greater than $3$?
0
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1answer
32 views

Prove tautology without truth using a truth table. [duplicate]

I am struggling to prove, without using truth tables, that the statement is a tautology. [(p→q)∧(q→r)]→(p→r) My work so far... ¬[(¬p∨q)∧(¬q∨r)]∨(¬p∨r) ...
0
votes
1answer
30 views

Using set theory to prove a function problem

I begin with: $$A = \{a \le x < x_0 | f(x) = 0 \}$$ $$B = \{x_0 < x \le b | f(x) = 0 \}$$ Let $c = \sup A$ and let $d = \sup B$ First to prove $f(x) > 0$ for $x \in (c, d)$ I will ...
1
vote
2answers
51 views

How to get to $5^3 \geq n^3$ in the proof by contradiction?

This is the same problem asked here. - Next step to take to reach the contradiction? Here is it again. I understand the solution - how you want to get to the fact 100 divides n^2 and then go ...
2
votes
2answers
23 views

Homology groups of orientable surfaces.

I am trying to show that the second (simplicial) homology group or an orientable surface is ismormophic to $\mathbb Z$. I can show that this group is non-trivial by triangulating the surface, and ...
0
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2answers
36 views

Prove that $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0})] = 0 \Rightarrow \lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$

Prove: $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0})] = 0 \Rightarrow \lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$ Proof: $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$ ...
1
vote
4answers
57 views

The divisibility of $a^p-1$ by $a-1$ and by $(a-1)^2$

I was working my way through some number theoretic proofs and being a newbie am stuck on this problem : Let a $\geq$ 2 and p be any positive integers , then prove that : $(a-1) \mid(a^p - ...
2
votes
2answers
40 views

Show that the set of all cofinite subsets of S is enumerable.

I've been having some trouble with this question. In fact, I spend a long time on a solution which I came to realize the next day it was entirely wrong. I feel completely stumped, and I could really ...
1
vote
3answers
33 views

Proving by contradiction that if $a\in\mathbb Q,b\in \mathbb R\setminus \mathbb Q$ then $a+b\in \mathbb R \setminus \mathbb Q$

I'm trying to prove by contradiction that if $a\in\mathbb Q,b\in \mathbb R\setminus \mathbb Q$ then $a+b\in \mathbb R \setminus \mathbb Q$, I already proved it with contra position and a direct proof ...
7
votes
0answers
71 views

Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

QN: What functions (from non-negative integers to non-negative integers) satisfy the condition $$f(f(f(n))) = f(n+1) + 1$$ Comment: Evidently $f(n) = n+ 1$ is one solution. Equally evidently no ...
3
votes
1answer
23 views

How to proceed with the following integration?

If $n$ is a positive integer, show that $$ \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin(t^2) dt = \frac{(-1)^n}{c}$$ for some $c \in [\sqrt{n\pi}, \sqrt{(n+1)\pi}]$ I have an idea that i can use Mean ...
1
vote
1answer
29 views

How can I prove this about the tangent line formula??

The equation of a tangent line to $f(x)$ at $x = t$ is $y = f'(t)(x - t) + f(t)$. Recently, I heard that it is also determined by the remainder of polynomial division of $f(x)$ by $(x-t)^2$. For ...
1
vote
1answer
36 views

For how many integers is this a prime number?

For how many integers $n$ is: $9 - (n-2)^2$ a prime number? I want to try this using a rigorous definition of prime number/ actual problem rather than try-error? Please only give hints, so I can do ...
0
votes
1answer
38 views

Proving an “OR” statement

If one wants to proof $P\vee Q$, is it sufficient to proof $\lnot P \rightarrow Q$? Because it makes intuitively more sense to me that $P\vee Q$ would be logically equivalent with $(\lnot P ...
0
votes
1answer
37 views

For what natural number $n$ is the following inequality true: $2^n \geq 2\cdot n^2$?

Can you solve this by using induction? The inequality is true for $n = 1$, but is false until $n = 7$. After the induction step I got $$2^n \geq n^2 + 2n + 1.$$ If you take the limit as $n$ ...
0
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1answer
24 views

Prove that the difference between two rational numbers is rational

This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt: Let $a$ ...
-1
votes
1answer
21 views

question on proving inequalities [on hold]

If I need to prove $t(x) \ge0 $, for all $ x>0$ and I prove that $t(x) \gt 0 $, for all $ x>0$ does that make for a proof or is it wrong?
0
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2answers
30 views

Olympiad minimum question, minimal value

If the numbers $A, B, C$ are such that the expression $\sqrt{A-B} + \sqrt{(B+3)^2} + C^2 - 4C + 4$ is as small as possible, then $A+B+C$ is? I thought start with, $A > B > C$ without loss of ...
0
votes
1answer
62 views

What is the contraposive of this statement?

I have to prove the negation of this statement: $$\forall a,b,c\in\mathbb{Z}{\;if\;a\;|\;b\} $$ But the fact that there is a "and" is very disturbing. I think that I am missing something because my ...
3
votes
3answers
60 views

Rank in row echelon form

$$A= \begin{bmatrix} a & 1 & a & 0 & 0 & 0 \\ 0 & b & 1 & b & 0 & 0 \\ 0 & 0 & c & 1 & c & 0 \\ 0 & 0 & 0 & d & 1 ...
0
votes
2answers
43 views

Matrices that commute with all matrices [duplicate]

Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \ \lambda \in \mathbb R\}$$ ($I_n$ is the $n \times n$ identity ...
1
vote
1answer
51 views

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: ...
0
votes
1answer
27 views

How to prove that the infinity norm of a matrix is the max of row sum?

I know how to prove that the 1-norm of a matrix is the max of the column sum, but not sure how to prove that the inf-norm is the max of the row sum. Any suggestion? Thanks
1
vote
2answers
101 views

Proving a function is onto?

Let $f: \mathbb{R}\setminus \{3\} \to \mathbb{R}\setminus \{1\}$ be defined by $f(x)=\dfrac{x+3}{x-3}$ Prove that $f$ is onto: Okay, here is the deal. I just started my first abstract algebra ...
-1
votes
1answer
16 views

Prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|))

For a,c, and x in the reals, prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|)). I'm trying to practice these kinds of questions, and any help or suggestions are greatly ...
0
votes
1answer
110 views

proof: $\sum\limits_{i=k}^n\binom{i}{k}=\binom{n+1}{k+1}$

Let $n ≥ 0$ and $k ≥ 0$ be integers. 1) How many bitstrings of length $n + 1$ have exactly $k + 1$ many $1$s? 2) Let $i$ be an integer with $k ≤ i ≤ n$. What is the number of bitstrings of length $n ...
-1
votes
2answers
31 views

Prove that if $y>1$, then $\forall M\in\mathbb{R}$, there exists an $N$ in the natural numbers s.t. $n\geq N$ implies $y^n>M$. [on hold]

For $y\in\mathbb{R}$, prove that if $y>1$, then $\forall M\in\mathbb{R}$, $\exists N\in\mathbb{N}$ such that $$ n≥N \implies y^n>M. $$ I'm not used to proving these kinds of questions so any ...
3
votes
1answer
64 views

Could someone take a crack at this number theory problem?

The question is stated as follows: If $\mathrm{gcd}(a,m)=1$ and $X$ is a complete residue system $\bmod m$, then the set obtained by multiplying each member of $X$ by $a$ is also a complete residue ...
1
vote
1answer
30 views

Define $x^3$ = $x \times x^2$. Prove that if $x_1, x_2$, . . . represents $x$, then $x_1^3$, $x_2^3$, . . . represents $x^3$ [on hold]

I'm a little bit lost on where to start this problem. My initial thought is to work backwards. Say $x^3$ is a Cauchy sequence. Then for some $j, k \geq m$ contingent on $n$, we have |${x^3}_j - ...