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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1answer
37 views

How would one derive conclusions from this?

Using only the 18 rules of inference without CP or IP derive the conclusion (x)(Ax ⊃ Bx) (x)(Bx ⊃ Cx) / (x)(Ax ⊃ Cx) As well as when using an Existential Quantifier (x)(Bx ⊃ Cx) (∃x)(Ax & ...
12
votes
3answers
191 views

Prove that $\frac{(p^{n}-1)(p^{n}-p)…(p^{n}-p^{n-1})}{n!} \in \mathbb{N}$ with $p$ a prime number and $n \in \mathbb{N}$

Apparently this question requires a method linked with linear algebra but I was wondering if it was possible to solve it in a formal way like an induction on $n$ or by using an identity for $p^{n}-1$ ...
1
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1answer
30 views

Using CP prove the truth of a tautology

Having trouble figuring out this tautology using CP and the rules of infrence [P ⊃ (Q ⊃ R)] ≡ [Q ⊃ (P ⊃ R)]
1
vote
1answer
44 views

Semantic tableau software

Is it possible to find software to perform semantic tableaus (as described in http://en.wikipedia.org/wiki/Method_of_analytic_tableaux) automatically? Right now I am proofing it by hand.
2
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3answers
46 views

Compare inequalities in a proof by induction

I am solving a proof by induction example. But I ended up with my hypothesis $$ a_{n-1} \geq \frac{2^n}{2}+n^2-2n+1 $$ and my inductive step $$ a_{n-1} \geq \frac{2^n}{2}+\frac{n^2}{2}-\frac{n}{2}. ...
1
vote
1answer
46 views

Consider the function $g(x)=xe^x$. Make and prove a conjecture about the $n^\text{th}$ derivative of $g$. [closed]

Please help on this homework problem I have in my Proofs class. My prof is really bad at explaining and I don't know how to answer this problem! Thank you!
0
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3answers
382 views

Prove that a rational number minus an irrational number must be irrational. [duplicate]

Please help with this homework problem I have! I don't know how to prove this.
0
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3answers
91 views

proof about limits of functions

Let $f:\mathbb R \to \mathbb R$ be such that $f(x), f'(x) and f''(x)$ are all positive for each $x \in \mathbb R$. Show that $\lim_{x \to \infty} f(x)=\infty$. So $f''(x)$ is the second derivative of ...
5
votes
1answer
77 views

Convergence of sequence of $L^{p}$ function

Given that $\Omega \subset \mathbb{R}^{n}$ is bounded. If you are given that $u_{k} \rightarrow u$ in $L^{p- \epsilon}(\Omega)$ and a functions $f: \mathbb{R} \rightarrow \mathbb{R}$ where ...
0
votes
1answer
37 views

Can you use proof by contradiction inside simple induction?

During a proof using simple induction, I assumed P(k) is true. Now in order to show P(k+1) is true using P(k), can I do a proof by contradiction on P(k+1) and say P(k) would be wrong if P(k+1) is ...
4
votes
3answers
132 views

A concave positive function on $[1,\infty)$ is uniformly continuous

Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
1
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1answer
52 views

Vector analysis - Curl of vector

How to prove it? I have tried several times to solve it, but I still get stuck everytime.
3
votes
2answers
89 views

Integer inequality: $x + y +z> a + b + c$ does not imply $xyz > abc$

Prove by contradiction that for any integers $x,y,z,a,b,c$ greater than $0$ such that $x+y>a+b$, it is not implied that $x\cdot y\cdot z>a\cdot b\cdot c$? Obviously this statement is true. ...
-2
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5answers
277 views

Basic set theory proof about cardinality of cartesian product of two finite sets

I'm really lost on how to do this proof: If $S$ and $T$ are finite sets, show that $|S\times T| = |S|\times |T|$. (where $|S|$ denotes the number of elements in the set) I understand why it is true, ...
2
votes
2answers
50 views

Alternate way to Prove or disprove $6\mid n(n+1)(n+2)$

This is my proof, I'm wondering if I'm correct, and how to do without induction. My Work Basis Step $$\frac{(1)(2)(3)}{6} = 1$$ Inductive Hypothesis Assume that $\dfrac{k(k+1)(k+2)}{6} = d$ where ...
1
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1answer
172 views

proving gradient of a function is always perpendicular to the contour lines

Can someone give an explanation of how such a proof would go, given a function example: $y = f(x)$
0
votes
1answer
30 views

Find number of all $a \in G $ such that $o(a) =3$

Let $G$ be a group and $|G|= 51$ find number of all $a \in G$ such that $o(a)=3$ My solution : by this theorem : if $|G|=pq$ that $ p ,q$ are prime. If $ q\nmid p-1 $ then $\quad$ $G \cong \Bbb ...
2
votes
1answer
51 views

Maximum load when placing N balls in N bins

In an academic paper I am reading the following.. When $n$ balls are placed into $n$ bins (each ball being placed into a bin chosen independently and uniformly at random) with high probability, the ...
0
votes
2answers
52 views

Proof for consecutive integers

Prove that if $n$ is an odd integer, $n^3$ is the sum of $n$ consecutive integers. I'm confused on how to prove something with consecutive integers.
0
votes
0answers
28 views

$P+Q:=\varphi_{O}^{-1}\left(\varphi_{O}(P)+\varphi_{O}(Q)\right)$

let X is affine space and $\overrightarrow{X}$ is vector space associted to X $$\begin{array}{ccccc} & \varphi_{O} : & X & \longrightarrow & \overrightarrow{X}\\ & & ...
0
votes
4answers
47 views

Induction Proof with a $\neq$ 1 [duplicate]

For $a \neq 1$ and $n>0$, $(1-a^{n+1})/(1-a)=1+a+a^2+...+a^n$ How do you prove this by induction?
1
vote
1answer
42 views

If $f(z)$ is an entire function, prove that it has a zero at $z_0$ of order $k\ge 1$ iff $z_0$ is a simple pole of $\frac{f'(z)}{f(z)}$

Let $f(z)$ be an entire function. Prove that $f(z)$ has a zero at $z_0$ of order $k\ge 1$ iff $z_0$ is a simple pole of $\frac{f'(z)}{f(z)}$ and the residue of $\frac{f'(z)}{f(z)}$ at $z_0$ is $k$. ...
2
votes
1answer
21 views

Prove an x exists with f(x) = f(x + T/2)

Suppose $f: \mathbb{R} → \mathbb{R}$ is a continuous and periodic function with period $T > 0$. Show that: there exists an $x \in \mathbb{R}$ such that $f(x) = f(x + T/2)$. We figured out we ...
1
vote
1answer
19 views

Procedure in proving inequalities (or bounds) involving minus sign

Usually when we want to bound an expression involving sums, it is easy to proceed by bounding each term separately since we "do not lose" too much using the triangle inequality $|x+y|\leq |x|+|y|$ ...
0
votes
2answers
59 views

L'Hôpital's rule in proofs

I was asked to prove that $\lim\limits_{x\to\infty}\frac{x^n}{a^x}=0$ when $n$ is some natural number and $a>1$. However, taking second and third derivatives according to L'Hôpital's rule didn't ...
0
votes
3answers
53 views

Prove/disprove: If $n\in \mathbb N$ with $n>2$ not prime, then $2n + 13$ is not prime.

Prove/disprove: If $n\in \mathbb N$ with $n>2$ not prime, then $2n + 13$ is not prime. From the context in which this question was set, I believe I have to prove/disprove it using ...
0
votes
2answers
38 views

Symmetric matrices and orthogonality

I'm struggling to make any progress with this question. I have defined C as the standard n-dimensional identity matrix. As A is semidefinite, I believe the diagonal matrix D must have positive ...
2
votes
0answers
58 views

Alternative proof of de l'Hospital theorem

I see that the usual proof of de l'Hospital theorem involves the mean value theorem and is carried on in a case-by-case fashion. Could you point out some other proofs of de l'Hospital that are ...
0
votes
3answers
93 views

Proof for nonexistent limit of $1/x$

Prove that: $$\lim_{x \to 0} \frac{1}{x}$$ Is non existent. This is my attempt: Assume The limit $= L$ Some real number $L$ By the definition of one sided limits we get from right and left ...
0
votes
4answers
41 views

proof of summation using $\displaystyle \binom{n}{r}$

Prove that $\sum_{r=0}^{n} \binom{n}{r}2^r = 3^n$ for $n \in \mathbb P$. "Hint: give me an argument having to do with the number of strings of length $n$ with $3$ symbols."
0
votes
2answers
60 views

Prove that $ (A \cup B) \cap C \subseteq A \cup (B \cap C)$

I'm trying to practice proof writing, and found the following text question: For all sets A,B,C: $ (A \cup B) \cap C \subseteq A \cup (B \cap C)$ The first step I was thinking of showing is that: ...
0
votes
3answers
27 views

Prove $\sqrt{a_n} \rightarrow \sqrt{L}$

If $(a_n) \rightarrow L, a_n > 0, \forall n \in \mathrm N, and L > 0$, then how can I show $\sqrt{a_n} \rightarrow \sqrt{L}$? I am given this hint, but I am not sure what to do with it: ...
1
vote
1answer
33 views

If (an)→ L, an > 0 for all n ∈ N, and L > 0, then prove that √an → √L .

To be honest, I don't even understand what the question is asking, and have no idea how to answer it. Any guidance would be great. I know convergent/divergent definitions, as well as basic limit laws, ...
0
votes
1answer
49 views

Solve this logical inference

I have the logic inference: Hypotheses: $A \implies (B \lor C)$ $A \lor (D \land B)$ Conclusion: $D \implies C$ I have these equivalence formations: Hypotheses: $A \lor (D \land B)$ $\lnot D ...
0
votes
2answers
35 views

Prove $\lim_{n \to \infty}ka_n = k\lim_{n \to \infty} a_n$

So I am given this to prove: $$\lim_{n \to \infty}ka_n = k\lim_{n \to \infty} a_n$$ This problem seems trivial to solve, but bear with me here. Would I be able to solve it by using the definition of ...
2
votes
1answer
33 views

Trouble Understanding Proof About Polynomials

In the question I have to prove that: There is no polynomial $P (x) = a_n x^n + a_{n−1}x^{n−1} + · · · + a_0$ with integer coefficients and of degree at least 1 with the property that $P(0), ...
2
votes
3answers
47 views

Prove that the sequence $\left\{\frac{n^2+1}{n+2}\right\}_{n=1}^\infty$ diverges to $\infty$

How would I be able to show that the sequence $\left\{\frac{n^2+1}{n+2}\right\}_{n=1}^\infty$ diverges to $\infty$? Here is what I do know: Suppose $M>0$ is given. We must find $N \in \mathrm N$ ...
2
votes
3answers
220 views

Prove that the sequence $\cos(n\pi/3)$ does not converge

EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge... $n=1$ to infinity of course. So, I have a bit of trouble ...
2
votes
0answers
78 views

MVT and functions

Let $f$ be defined on an open interval $I := (a,b)$. (a) Let $x$ and $y$ be real numbers such that $x<y$. Show that if $z \in [x,y]$, then there is some $t \in [0,1]$ such that $z=tx+(1-t)y$. (b) ...
0
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1answer
20 views

How to show $[F(a):F[a^4+a^2+1]$ is finite.

I have an element $a$ in an extension field $F$. I'm asked if it is true that $a^4+a^2+1$ is algebraic over $F$ if and only if $a$ is algebraic over $F$. I know that if I can show ...
2
votes
0answers
31 views

How to simplify expression with Fermat's little theorem

I don't quite understand how to reduce (25^74 + 53^27)^(10) I thought I would reduce to (4^74 + 4^27)^10 and then (4^(7*10 + 4) + 4^ (7*3 + 6))^(7*1 + 3) And then (4^4 + 4^6)^3 but that doesn't ...
2
votes
1answer
36 views

$K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

Let $F$ be a field and $K$ be an extension field of $F$. Proof\Counterexample: $K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$. I haven't ...
1
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2answers
62 views

Express a Proposition In Formal Logic

I am doing a question where I have to express: There is no largest prime number, in formal logic. This is the solution given: Of course this is a true statement, so it could be expressed by the ...
8
votes
4answers
358 views

Unconventional (but instructive) proofs of basic theorems of calculus

Inspired by this questions asked on MathOverflow, I would like to ask if you know some "sophisticated" proofs of the basic theorems in a calculus course (that is, the ones that you can find, for ...
1
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2answers
97 views

Proof of dilogarithm reflection formula $\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$

How to prove $$\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$$ I havent started, any hints?
0
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1answer
33 views

How do I prove this by induction? [duplicate]

thank you for taking the time to help me with the question. I am struggling to use proof by induction for this formula: $$\sum_{k=0}^{n}k\times k! = (n + 1)! - 1$$ So far, I came up with: $$S(n) = ...
0
votes
3answers
30 views

For every integer m, 2 divides m, and 4 doesn't divide m, there are no integers, x and y that satisfy x^2 + 3y^2 = m.

For every integer m, 2 divides m, and 4 doesn't divide m, there are no integers, x and y that satisfy x^2 + 3y^2 = m. Use a contradiction (assume the negation is true) Is my negation of the ...
1
vote
1answer
85 views

If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$.

Let $a$ and $b$ be elements in extension field $F$. Is it true that: If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$? I just did the same ...
0
votes
0answers
22 views

Showing relation is transitive (a,b) $\in$ R iff 2|(a+b)

Let R be the relation on natural numbers defined by (a,b) $\in$ R iff 2|(a+b) show it is transitive.
1
vote
1answer
85 views

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$?

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$? I'm asked to give a proof or a counterexample. I'm a bit confused on the notation of $\mathbb R(a)$, what does this ...